Integer Sequences

Table of contents. An “*" indicates nice.  Number lists are here.

S000001    Number of primitive prime factors of 3^n - 1.

S000002    Number of primitive prime factors of 4^n - 1.

S000003    Number of primitive prime factors of 5^n - 1.

S000004    Number of primitive prime factors of 6^n - 1.

S000005    Number of primitive prime factors of 7^n - 1.

S000006    Number of primitive prime factors of 8^n - 1.

S000007    Number of primitive prime factors of 9^n - 1.

S000008    Number of primitive prime factors of 3^n + 1.

S000009    Number of primitive prime factors of 4^n + 1.

S000010    Number of primitive prime factors of 5^n + 1.

S000011    Number of primitive prime factors of 6^n + 1.

S000012    Number of primitive prime factors of 7^n + 1.

S000013    Number of primitive prime factors of 8^n + 1.

S000014    Number of primitive prime factors of 9^n + 1.

S000015    Number of primitive prime factors of 10^n + 1.

S000016    Numbers that are unique-period primes in bases 2 and 3.

S000017    Numbers that are unique-period primes in bases 2 and 4.

S000018    Numbers that are unique-period primes in bases 3 and 9.

S000019    Unique-period primes in base 3 in the order they are found.

S000020    Unique-period primes in base 4 in the order they are found.

S000021    Unique-period primes in base 5 in the order they are found.

S000022    Unique-period primes in base 6 in the order they are found.

S000023    Unique-period primes in base 7 in the order they are found.

S000024    Unique-period primes in base 8 in the order they are found.

S000025    Unique-period primes in base 9 in the order they are found.

S000026    Unique-period primes in base 3, sorted.

S000027    Unique-period primes in base 4, sorted.

S000028    Unique-period primes in base 5, sorted.

S000029    Unique-period primes in base 6, sorted.

S000030    Unique-period primes in base 7, sorted.

S000031    Unique-period primes in base 8, sorted.

S000032    Unique-period primes in base 9, sorted.

S000033    Unique-period primes in base 3, written in base 3.

S000034    Unique-period primes in base 4, written in base 4.

S000035    Unique-period primes in base 5, written in base 5.

S000036    Unique-period primes in base 6, written in base 6.

S000037    Unique-period primes in base 7, written in base 7.

S000038    Unique-period primes in base 8, written in base 8.

S000039    Unique-period primes in base 9, written in base 9.

S000040    Irregular table of primitive prime factors of 5^n - 1.

S000041    Irregular table of primitive prime factors of 6^n - 1.

S000042    Irregular table of primitive prime factors of 7^n - 1.

S000043    Irregular table of primitive prime factors of 8^n - 1.

S000044    Irregular table of primitive prime factors of 9^n - 1.

S000045    Irregular table of primitive prime factors of 3^n + 1.

S000046    Irregular table of primitive prime factors of 4^n + 1.

S000047    Irregular table of primitive prime factors of 5^n + 1.

S000048    Irregular table of primitive prime factors of 6^n + 1.

S000049    Irregular table of primitive prime factors of 7^n + 1.

S000050    Irregular table of primitive prime factors of 8^n + 1.

S000051    Irregular table of primitive prime factors of 9^n + 1.

S000052    Irregular table of primitive prime factors of 10^n + 1.

S000053    Number of 2's in the n X n multiplication table (mod n).

S000054    Composite numbers n that divide the sum of the composite numbers up to n.

S000055    Numbers n that divide the sum of the nonprime numbers up to n.

S000056    Numbers n that divide the sum of the composite numbers up to n.

S000057    Irregular table of conjectured Fibonacci numbers with exactly n bits set in binary.

S000058    Lucas numbers written in binary notation.

S000059    Irregular table of conjectured Lucas numbers with exactly n bits set in binary.

S000060    Conjectured number of Lucas numbers with exactly n bits set in their binary representation.

S000061*  Symmetric n x n matrix of binomial coefficients.

S000062*  Inverse of the symmetric n x n matrix of binomial coefficients in S000061.

S000063    Permanent of the symmetric n x n matrix of binomial coefficients in S000061.

S000064*  Conjectured maximum number of primes between n*k and n*(k+1) for some k > 0.

S000065    Conjectured value of k for numbers in S000064.

S000066    Irregular triangle of the offsets used in S000064 and S000065.

S000067*  First number of 3 consecutive twin prime pairs whose first numbers differ by 18 or less.

S000068*  First number of 4 consecutive twin prime pairs whose first numbers differ by 30 or less.

S000069*  First number of 5 consecutive twin prime pairs whose first numbers differ by 36 or less.

S000070*  First number of 6 consecutive twin prime pairs whose first numbers differ by 48 or less.

S000071    Cases in a 1978 theorem of Ecklund, Eggleton, Erdos, and Selfridge.

S000072    Cases in a 1978 corollary of Ecklund, Eggleton, Erdos, and Selfridge.

S000073*  Minimum length of an interval containing n twin primes, measured mid-to-mid.

S000074    Minimum number of numbers in an interval containing n twin primes.

S000075    Primes of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for positive x, y, and z.

S000076    Primes not of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for positive x, y, and z.

S000077    Primes of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for nonnegative x, y, and z.

S000078    Primes not of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for nonnegative x, y, and z.

S000079    Numbers that are the difference of two perfect numbers.

S000080    Primes p at which (nextprime(p) - p) / (p - prevprime(p)) is a new record high.

S000081    Primes p at which (nextprime(p) - p) / (p - prevprime(p)) is a new record low.

S000082    Primes p at which (nextprime(p) - p) / (p - prevprime(p)) is a new record high or a record low.

S000083    Cumulative sum of the squares of nonprimes.

S000084    Cumulative sum of the cubes of nonprimes.

S000085    Cumulative sum of the 4-th powers of nonprimes.

S000086    Cumulative sum of the 5-th powers of nonprimes.

S000087    Cumulative sum of the squares of composite numbers.

S000088    Cumulative sum of the cubes of composite numbers.

S000089    Cumulative sum of the 4-th powers of composite numbers.

S000090    Cumulative sum of the 5-th powers of composite numbers.

S000091*  The matrix whose n-th row consists of increasing numbers whose largest prime factor is
                 prime(n).

S000092    Length of Lucas(n) when written in binary.

S000093    Numbers n such that x^2 + y^2 (mod n) assumes every number in Zn, where x and y are
                 nonzero.

S000094    Numbers n such that x^2 + y^2 (mod n) does not assume every number in Zn, where x and y
                 are nonzero.

S000095    Numbers n such that Fib(2n-1) and Fib(2n+1) are both prime.

S000096    Primes p such that p + 270 is also prime.

S000097    Table of initial primes such that there are k consecutive primes that are equal (mod n), with
                 k > 1 and n > 2.

S000098    Figurate primes, numbers of the form binomial(p^x,y), where p is prime and x and y are
                 positive integers.

S000099    Least k such that 10^n + k, 10^n + k + 2, and 10^n + k + 6 are prime.

S000100    Least k such that 10^n + k, 10^n + k + 4, and 10^n + k + 6 are prime.

S000101    Offset of the first prime triplet greater than 10^n; minimum of S000100(n) and S000101(n).

S000102    Numbers n such that Fib(n-1) and Fib(n+1) are both prime.

S000103    Numbers n such that Lucas(n-1) and Lucas(n+1) are both prime.

S000104*  Minimum length of an interval containing n twin primes.

S000105    First number of 7 consecutive twin prime pairs whose first numbers differ by 62 or less.

S000106    Number of different admissible twin-prime tuples.

S000107    Primes not of form x^2 + 11*y^2.

S000108    Primes not of form x^2 + 13*y^2.

S000109    Primes not of form x^2 + 14*y^2.

S000110    Primes not of form x^2 + 15*y^2.

S000111    Primes not of form x^2 + 17*y^2.

S000112    Primes not of form x^2 + 18*y^2.

S000113    Primes not of form x^2 + 19*y^2.

S000114    Discriminant of odd primitive forms of regular ternary forms.

S000115    Discriminant of even primitive forms of regular ternary forms.

S000116*  Discriminant of even and odd primitive forms of regular ternary forms, sorted.

S000117*  Discriminants and 6-term forms of even and odd primitive forms of regular ternary forms, sorted.

S000118*  Lists of four numbers (a,b,c,d) such that a*x^2 + b*y^2 + c*z^2 + d*u^2 represents all
                 nonnegative numbers.

S000119    The number of numbers i such that p = i (mod n) for some powerful number p and 0 <= i < n.

S000120    The number of numbers i such that p = i (mod n) for no powerful number p and 0 <= i < n.

S000121    Triangle of the least powerful number p such that p = i (mod n), or 0 if no such p exists.

S000122    Lists of four numbers (a,b,c,d) such that a*x^2 + b*y^2 + c*z^2 + d*u^2 represents all
                 nonnegative numbers, sorted.

S000123    Prefixes (first three terms) appearing in quadruples of S000118 and S000122.

S000124    Indices of Fibonacci numbers that do not have a factor of the form 4k+1.

S000125    Number of ways that n can be written as the sum of four positive nondecreasing squares and a prime.

S000126    Number of ways that n can be written as the sum of four nonnegative nondecreasing squares and a
                 prime.

S000127    Number of ways that n can be written as the sum of four positive squares and a prime.

S000128    Number of ways that n can be written as the sum of four nonnegative squares and a prime.

S000129    Number of ways that n can be written as the sum of three positive nondecreasing squares and a prime.

S000130    Number of ways that n can be written as the sum of three nonnegative nondecreasing squares and a
                 prime.

S000131    Number of ways that n can be written as the sum of three positive squares and a prime.

S000132    Number of ways that n can be written as the sum of three nonnegative squares and a prime.

S000133    Number of ways that n can be written as the sum of two positive nondecreasing squares and a prime.

S000134    Number of ways that n can be written as the sum of two nonnegative nondecreasing squares and a
                 prime.

S000135    Number of ways that n can be written as the sum of two positive squares and a prime.

S000136    Number of ways that n can be written as the sum of two nonnegative squares and a prime.

S000137    Table of triples (a,b,c) of quadratic forms having the number of discriminants given in A107628.

S000138*  Table of 4-tuples (d,a,b,c) of quadratic forms having the number of discriminants d given in A107628.

S000139    Negative of discriminants of quadratic forms in S000137.

S000140*  The numerator of sigma(n)/n for the superabundant numbers n.

S000141*  The denominator of sigma(n)/n for the superabundant numbers n.

S000142    Powers such that the interval to the next power contains exactly one prime.

S000143    Powers such that the interval to the previous power contains exactly one prime.

S000144    Powers such that the interval to the next power contains exactly two primes.

S000145    Powers such that the interval to the previous power contains exactly two primes.

S000146    Powers such that the interval to the next power contains exactly three primes.

S000147    Powers such that the interval to the previous power contains exactly three primes.

S000148    Powers such that the interval to the next power contains exactly four primes.

S000149    Powers such that the interval to the previous power contains exactly four primes.

S000150    Powers such that the interval to the next power contains exactly five primes.

S000151    Powers such that the interval to the previous power contains exactly five primes.

S000152    Primes p such that p + 246 is also prime.

S000153*  Sine: table in which the k-th row is round(sin(2*Pi*n/10^k)) for n = 0..10^k.

S000154    Cosine: table in which the k-th row is round(cos(2*Pi*n/10^k)) for n = 0..10^k.

S000155    Tangent: table in which the k-th row is round(tan((Pi/2)*n/10^k)) for n = 0..10^k - 1.

S000156    Cotangent: table in which the k-th row is round(cot((Pi/2)*n/10^k)) for n = 1..10^k.

S000157    Arcsine: table in which the k-th row is round(arcsin(n/10^k)) for n = -10^k..10^k.

S000158    Arccosine: table in which the k-th row is round(arccos(n/10^k)) for n = -10^k..10^k.

S000159    Nonnegative numbers n for which n^2 - n + 41 is not squarefree.

S000160    Nonnegative numbers n for which n^2 - n - 56 is not squarefree.

S000161    Nonnegative numbers n for which n^2 - n - 65 is not squarefree.

S000162    First differences of numbers n for which n^2 - n + 41 is not squarefree (S000159).

S000163    Number beginning n consecutive numbers that are not cube-free.

S000164    Number beginning n consecutive numbers that are not free of fourth powers.

S000165    The most common digit in the decimal form of 2^n; the least one if there are duplicates.

S000166    Number of times the digit S000165(n) appears in the decimal form of 2^n.

S000167    The digit in the longest run in the decimal form of 2^n; the least one if there are duplicates.

S000168    Longest run of the digit S000167(n) in the decimal form of 2^n.

S000169    The most common digit in the decimal form of 3^n; the least one if there are duplicates.

S000170    Number of times the digit S000169(n) appears in the decimal form of 3^n.

S000171    The digit in the longest run in the decimal form of 3^n; the least one if there are duplicates.

S000172    Longest run of the digit S000171(n) in the decimal form of 3^n.

S000173    The most common digit in the decimal form of 5^n; the least one if there are duplicates.

S000174    Number of times the digit S000173(n) appears in the decimal form of 5^n.

S000175    The digit in the longest run in the decimal form of 5^n; the least one if there are duplicates.

S000176    Longest run of the digit S000175(n) in the decimal form of 5^n.

S000177    The most common digit in the decimal form of 7^n; the least one if there are duplicates.

S000178    Number of times the digit S000177(n) appears in the decimal form of 7^n.

S000179    The digit in the longest run in the decimal form of 7^n; the least one if there are duplicates.

S000180    Longest run of the digit S000179(n) in the decimal form of 7^n.

S000181    Degrees of minimal polynomials that have a Salem number less than 1.3 as a zero.

S000182*  Number of polynomials of degree S000181(n) that have a Salem number less than 1.3 as a zero.

S000183*  Degrees of minimal polynomials having a Salem number under 1.3 as a root, sorted.

S000184    Coefficients of the minimal polynomial having a Salem number under 1.3 as a root, sorted.

S000185    Primitive Pythagorean triples (a, b, c^3) for prime c.

S000186    Primitive Pythagorean triples (a, b, c^3) for positive integer c.

S000187    Number of pairs of twin primes between e^n and e^(n+1) (where e = 2.718281828…).

S000188    Number of pairs twin primes between f^n and f^(n+1) (where f = e^2 = (2.718281828…)^2).

S000189    Numbers n such that e^n is next to a prime number.

S000190    Prime number s(n) such that (primepi(p) - primepi(s(n)) / (p - s(n)) is smaller for greater prime p.

S000191    The indices of the primes in S000190.

S000192    Floor of the n-th zero of the Bessel function J0.

S000193    Floor of the n-th zero of the Bessel function J1.

S000194    Floor of the n-th zero of the Bessel function J2.

S000195    Floor of the n-th zero of the Bessel function J3.

S000196    Floor of the n-th zero of the Bessel function J4.

S000197    Floor of the n-th zero of the Bessel function J5.

S000198    Floor of the n-th zero of the Bessel function Y0.

S000199    Floor of the n-th zero of the Bessel function Y1.

S000200    Floor of the n-th zero of the Bessel function Y2.

S000201    Floor of the n-th zero of the Bessel function Y3.

S000202    Floor of the n-th zero of the Bessel function Y4.

S000203    Floor of the n-th zero of the Bessel function Y5.

S000204    Floor of the n-th zero of the Airy function Ai.

S000205    The rounded n-th zero of the Airy function Ai.

S000206    Floor of the n-th zero of the Airy function Bi.

S000207    The rounded n-th zero of the Airy function Bi.

S000208*  For p = prime(n), the total of the absolute value of the differences of the inverses of the 
                 numbers 1..p-1 (mod p).

S000209    Maximum difference between primes less than 2^n.

S000210    Maximum difference between primes less than e^n.

S000211    Numbers n such that w(n) = sum_{k=0..n} binomial(n,k) binomial(n+k,k)/(2*k-1) is prime.

S000212    Primes w(n) for the n in S000211.

S000213    Sum of the reciprocals of Catalan numbers (A000108).

S000214    Alternating sum of the reciprocals of Catalan numbers (A000108).

S000215    Sum of the reciprocals of even-index Catalan numbers (A048990).

S000216    Sum of the reciprocals of odd-index Catalan numbers (A024492).

S000217    Index of Fibonacci numbers not containing the number 1.

S000218    Index of Fibonacci numbers not containing the number 2.

S000219    Index of Fibonacci numbers not containing the number 3.

S000220    Index of Fibonacci numbers not containing the number 4.

S000221    Index of Fibonacci numbers not containing the number 6.

S000222    Index of Fibonacci numbers not containing the number 7.

S000223    Index of Fibonacci numbers not containing the number 8.

S000224    Index of Fibonacci numbers not containing the number 9.

S000225    Index of Lucas numbers not containing the number 0.

S000226    Index of Lucas numbers not containing the number 1.

S000227    Index of Lucas numbers not containing the number 2.

S000228    Index of Lucas numbers not containing the number 3.

S000229    Index of Lucas numbers not containing the number 4.

S000230    Index of Lucas numbers not containing the number 5.

S000231    Index of Lucas numbers not containing the number 6.

S000232    Index of Lucas numbers not containing the number 7.

S000233    Index of Lucas numbers not containing the number 8.

S000234    Index of Lucas numbers not containing the number 9.

S000235    Number of composite numbers less than prime(n)^2.

S000236*  Pi/4 - 1/2.

S000237    Pi/3 - sqrt(3)/4.

S000238    Pi/6 - sqrt(3)/4.

S000239    Three periodic orbits in the 3x-1 iteration.

S000240*  Length of the 3x-1 iteration applied to odd numbers.

S000241    Where records occur in the 3x-1 iteration.

S000242    The record values of the length of the 3x-1 iteration.

S000243    Pi/5 - sqrt((5 + sqrt(5))/32).

S000244    Pi/7 - sqrt((1 + sin(Pi/14))/8).

S000245    Pi/8 - sqrt(2)/4.

S000246    Pi/9 - sqrt((1 - sin(Pi/18))/8).

S000247    Pi/10 - sqrt((5 - sqrt(5))/32).

S000248    Pi/11 - sqrt((1 - sin(3*Pi/22))/8).

S000249*  Pi/12 - 1/4.

S000250    Floor of 1000000 times the area of one circular segment outside the unit n-gon.

S000251    Floor of 1000000 times the area of the n circular segments outside the unit n-gon.

S000252    Length of the reduced 3x-1 iteration applied to odd numbers.

S000253    Where records occur in the reduced 3x-1 iteration.

S000254    The record values of the length of the reduced 3x-1 iteration.

S000255    The number of primes up to n divides the n-th Fibonacci number.

S000256*  Numbers n such that the n-th prime divides the n-th Fibonacci number.

S000257    The S000256(n)-th prime.

S000258    Dropping times of odd numbers in the 3x+1 problem (or the Collatz problem).

S000259    Dropping times of numbers 3+4*n in the 3x+1 problem (or the Collatz problem).

S000260*  Irregular triangle of a conjectured periodicity in the Collatz (3x+1) iteration.

S000261    Dropping times in the 3x+1 (Collatz) iteration of 2^n-1.

S000262    Stopping times in the 3x+1 (Collatz) iteration of 2^n-1.

S000263    Irregular triangle of numbers x such that primepi(n * x) = n + x.

S000264    Number of numbers x such that primepi(n * x) = n + x.

S000265    Last number x such that primepi(n * x) = n + x.

S000266    Least k such that 2^n - k - 2 and 2^n - k are twin primes.

S000267    Least k such that 2^n - k and 2^n - k + 2 are twin primes.

S000268*  Irregular triangle of numbers that have the same length Collatz (3x+1) iteration.

S000269*  Dropping patterns of the 3x+1 (Collatz) iteration.

S000270    All numbers n that solve x^2 = 57 * 2^n + 117440512.

S000271    All numbers n that solve x^2 = 165 * 2^n + 26404.

S000272    The last number (or 0 if there is none) in row n of the irregular triangle A177789 (Collatz related).

S000273    The nonzero terms of S000272.

S000274    Records in the lengths of the Collatz (3x+1) iteration.

S000275    Least integer c such that binomial(c,n) has n prime factors greater than n.

S000276    Smallest number k > 0 such that the interval [k^2,(k+1)^2] contains n pairs of twin primes.

S000277    Numbers beginning 23 consecutive numbers whose squares sum to a square.

S000278    Numbers beginning 26 consecutive numbers whose squares sum to a square.

S000279    Numbers beginning 33 consecutive numbers whose squares sum to a square.

S000280    Numbers beginning 47 consecutive numbers whose squares sum to a square.

S000281    Numbers beginning 50 consecutive numbers whose squares sum to a square.

S000282    Numbers beginning 59 consecutive numbers whose squares sum to a square.

S000283    Numbers n (excluding squares > 1) such that sum of squares of n consecutive integers >= 1 is a square.

S000284    Numbers beginning 96 consecutive numbers whose squares sum to a square.

S000285    Numbers beginning 338 consecutive numbers whose squares sum to a square.

S000286    Position of first zero digit in 2^n, counting from the right.

S000287    Position of first zero digit in 3^n, counting from the right.

S000288    Position of first zero digit in 4^n, counting from the right.

S000289    Position of first zero digit in 5^n, counting from the right.

S000290    Position of first zero digit in 6^n, counting from the right.

S000291    Position of first zero digit in 7^n, counting from the right.

S000292    Position of first zero digit in 8^n, counting from the right.

S000293    Position of first zero digit in 9^n, counting from the right.

S000294    Irregular triangle of numbers n such that 2^n does not contain 0, 1, …, 9.

S000295    Irregular triangle of numbers n such that 3^n does not contain 0, 1, …, 9.

S000296    Irregular triangle of numbers n such that 4^n does not contain 0, 1, …, 9.

S000297    Irregular triangle of numbers n such that 5^n does not contain 0, 1, …, 9.

S000298    Irregular triangle of numbers n such that 6^n does not contain 0, 1, …, 9.

S000299    Irregular triangle of numbers n such that 7^n does not contain 0, 1, …, 9.

S000300    Irregular triangle of numbers n such that 8^n does not contain 0, 1, …, 9.

S000301    Irregular triangle of numbers n such that 9^n does not contain 0, 1, …, 9.

S000302    2^n - S000272(n).

S000303    Sequence S000302 without the zeros.

S000304    A chain of twin primes.

S000305*  Another chain of twin primes.

S000306    Ratio (1+b(n+1)) / b(n) for the sequence b = S000304.

S000307    Factors used in forming the lower twin primes in S000305.

S000308    First prime of 9 primes in a range of 31 numbers.

S000309    First prime of 7 primes in a range of 21 numbers.

S000310    Least non-border binomial coefficient having n prime factors, or 0 if none exists.

S000311    Least row in the table of binomial coefficients having a number with n distinct prime factors.

S000312    For the n-th twin primes x±1, the least number k*x-1 such that k*x±1 are twin primes for k > 1.

S000313    For the n-th twin primes x±1, the least number k > 1 such that k*x±1 are twin primes.

S000314    Prime values of m^2 - 2*n^2 for prime m and n.

S000315    Numbers n such that 2n+1 and 2n+3 are prime.

S000316    Numbers n such that 2n+1, 2n+3, and 2n+7 are prime.

S000317    Numbers n such that 2n+1, 2n+3, 2n+7, and 2n+9 are prime.

S000318    Numbers n such that 2n+1, 2n+3, 2n+7, 2n+9, and 2n+13 are prime.

S000319    Numbers n such that 2n+5, 2n+7, and 2n+11 are prime.

S000320    Numbers n such that 2n+5, 2n+7, 2n+11, and 2n+13 are prime.

S000321    Numbers n such that 2n+5, 2n+7, 2n+11, 2n+13, and 2n+17 are prime.

S000322*  Pairs of prime numbers {p1, p2} that list the range that can generate primes, as in S000321.

S000323    The index of Lucas numbers that are not squarefree.

S000324    Lucas numbers that are not squarefree.

S000325    The squarefull part of the n-th Lucas number.

S000326    The squarefull part of the n-th Fibonacci number.

S000327    Primes having numbers with 4 and 2 prime factors (counting multiplicity) before and after them.

S000328    Primes having numbers with 2 and 4 prime factors (counting multiplicity) before and after them.

S000329    Primes having numbers with 2 and 4 prime factors (counting multiplicity) before or after them.

S000330    Primes having numbers with a total of 6 prime factors (counting multiplicity) before or after them.

S000331    Records of the total number of prime factors (counted multiply) in the numbers between consecutive primes.

S000332    The first number in the pair shown in S000331.

S000333    The second number in the pair shown in S000331.

S000334    Records of the total number of prime factors in the numbers between consecutive primes.

S000335    The first number in the pair shown in S000334.

S000336    The second number in the pair shown in S000334.

S000337*  Hypotenuses of right triangles having legs that are triangular numbers.

S000338*  Three sides of right triangles having legs that are triangular numbers, ordered by hypotenuse.

S000339    Reduced hypotenuses of right triangles in which the hypotenuse and one leg are triangular numbers.

S000340*  Three sides of right triangles having the hypotenuse and one leg that are triangular numbers.

S000341    Number of solutions to 4/prime(n) = 1/x + 1/y + 1/z for integers x, y, z with 0 < x < y < z.

S000342    Primes at which maximums occur in the Erdos-Straus conjecture graph (see S000341).

S000343    The maximums of the Erdos-Straus conjecture graph at the primes in S000342.

S000344    Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 2.

S000345    Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 3.

S000346    Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 4.

S000347    Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 5.

S000348    Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 6.

S000349    Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 7.

S000350    Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 8.

S000351    Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 9.

S000352    Palindromic triangular numbers in base 2.

S000353    Palindromic triangular numbers in base 3.

S000354    Palindromic triangular numbers in base 4.

S000355    Palindromic triangular numbers in base 5.

S000356    Palindromic triangular numbers in base 6.

S000357    Palindromic triangular numbers in base 7.

S000358    Palindromic triangular numbers in base 8.

S000359    Palindromic triangular numbers in base 9.

S000360    Numbers n such that 3^n-1 has only one primitive prime factor.

S000361    Numbers n such that 4^n-1 has only one primitive prime factor.

S000362    Numbers n such that 5^n-1 has only one primitive prime factor.

S000363    Numbers n such that 6^n-1 has only one primitive prime factor.

S000364    Numbers n such that 7^n-1 has only one primitive prime factor.

S000365    Numbers n such that 8^n-1 has only one primitive prime factor.

S000366    Numbers n such that 9^n-1 has only one primitive prime factor.

S000367    Numbers n such that 2^n+1 has only one primitive prime factor.

S000368    Numbers n such that 3^n+1 has only one primitive prime factor.

S000369    Numbers n such that 4^n+1 has only one primitive prime factor.

S000370    Numbers n such that 5^n+1 has only one primitive prime factor.

S000371    Numbers n such that 6^n+1 has only one primitive prime factor.

S000372    Numbers n such that 7^n+1 has only one primitive prime factor.

S000373    Numbers n such that 8^n+1 has only one primitive prime factor.

S000374    Numbers n such that 9^n+1 has only one primitive prime factor.

S000375    Numbers n such that 10^n+1 has only one primitive prime factor.

S000376    Irregular array of positive numbers that form palindromic tetrahedral numbers in bases 2 to 9.

S000377    Irregular array of positive palindromic tetrahedral numbers in bases 2 to 9.

S000378    (3^(4*n - 1) - 1)/2.

S000379    For the n in S000378, the tetrahedral number n*(n+1)*(n+2)/6 in base 9.

S000380    Irregular array of positive numbers that form palindromic square pyramidal numbers in bases 2 to 10.

S000381    Irregular array of positive palindromic square pyramidal numbers in bases 2 to 10.

S000382    Deteminant of the n x n symmetric matrix whose lower triangular part is the binomial numbers mod 2.

S000383    Let a(n) = 2^n + a(n-2) for n > 1, a(0) = 3, and a(1) = 3.

S000384    Numbers n such that the number of composite numbers around n^2 is a new record.

S000385    The record number of composite numbers around S000384(n)^2.

S000386    Number of solutions to 4/prime(n) = 1/x + 1/y + 1/z for integers x, y, z with 0 < x <= y <= z.

S000387    The practical part of positive integer n.

S000388    Numbers n for which a sum of n-th roots of unity is a new nonzero minimum (in absolute value).

S000389    Gaussian integers near the curve z = exp(I * t).

S000390    Gaussian primes near the curve z = exp(I * t).

S000391*  Numbers n such that Gaussian primes x + I*n appear to have distributions similar to 4m+3 primes.

S000392    An adjacency matrix for the 10-node Petersen graph.

S000393    Eigenvalues of the adjacency matrix for the 10-node Petersen graph.

S000394    Palindromic primes in base 17, but written here in base 10.

S000395    Palindromic primes in base 18, but written here in base 10.

S000396    Palindromic primes in base 19, but written here in base 10.

S000397    Palindromic primes in base 20, but written here in base 10.

S000398    Palindromic primes in base 21, but written here in base 10.

S000399    Palindromic primes in base 22, but written here in base 10.

S000400    Palindromic primes in base 23, but written here in base 10.

S000401    Palindromic primes in base 24, but written here in base 10.

S000402    Palindromic primes in base 25, but written here in base 10.

S000403    The n-th palindromic prime in base n, but written here in base 10.

S000404    Triangle of the number of prime factors (not counted multiply) of first-quadrant Guassian integers.

S000405    Triangle of the number of prime factors (counted multiply) of first-quadrant Guassian integers.

S000406    Triangle of the number of divisors of first-quadrant Guassian integers.

S000407    The first Gaussian integer having n distinct prime factors in the triangle S000404.

S000408    Diagonal having the first Gaussian integer with n distinct prime factors in the triangle S000404.

S000409    Gaussian primes in the first quadrant, sorted by magnitude and real part.

S000410    Gaussian primes strictly in the first quadrant, sorted by magnitude and real part.

S000411    Gaussian primes strictly in the first quadrant, sorted by magnitude and real part < imaginary part.

S000412    Gaussian primes strictly in the first quadrant, sorted by magnitude and imaginary part < real part.

S000413    Gaussian primes strictly in the first quadrant, sorted by magnitude and imaginary part.

S000414    Gaussian primes in the first quadrant, sorted by magnitude and imaginary part.

S000415*  The Gaussian primes, sorted by magnitude and angle.

S000416    Distance squared to the center of a 4-lion from the origin.

S000417    Position of the center of a 4-lion in the first quadrant, sorted by magnitude and real part.

S000418    Position of the center of a 4-lion in the octant x >= y >= 0.

S000419    Position of the center of a 4-lion in the octant y >= x >= 0.

S000420    Position of the center of a 4-lion in the first quadrant, sorted by magnitude and imaginary part.

S000421    Distance squared to the center of a “real" 4-lion from the origin.

S000422    Position of the center of a “real” 4-lion in the first quadrant, sorted by magnitude and real part.

S000423    Position of the center of a “real" 4-lion in the octant x >= y >= 0.

S000424    Position of the center of a “real" 4-lion in the octant y >= x >= 0.

S000425    Position of the center of a “real" 4-lion in the first quadrant, sorted by magnitude and imaginary part.

S000426*  Sum of the absolute values of the coefficients of Phi(n,x), the n-th cyclotomic polynomial.

S000427*  Sum of the squares of the coefficients of Phi(n,x), the n-th cyclotomic polynomial.

S000428*  The Bombieri norm of Phi(n,x), the n-th cyclotomic polynomial.

S000429    The triangle of nonzero (m - 1) * (n - 1) for 1 < m < n.

S000430    The triangle of nonzero (m - 1) * (n - 1) - 1 for 1 < m < n.

S000431    Three-dimensional triangle of the conductor of a, b and c for the expression a*x + b*y + c*z.

S000432    Three-dimensional triangle of one less than the conductor of a, b, c for the expression a*x + b*y + c*z.

S000433    Conductor of prime(n), prime(n+1), and prime(n+2).

S000434    Conjectured value of the conductor of prime(n), prime(n+1), prime(n+2),….

S000435*  Conjectued number of terms required for the conductor function in S000434 to converge.

S000436    Primes p such that the iteration in S000334 converges after more than primepi(p) steps.

S000437    Primes p such that the iteration in S000334 converges at or before primepi(p) steps.

S000438*  Triangle of the power tower k^k^k^... (mod n) for k = 1..(n-1).

S000439    The height of the exponents in S000438.

S000440    Number of odd primes in Collatz sequences having only 1, even numbers, and primes.

S000441*  The first prime that produces exactly n odd primes in the Collatz iteration; the other terms are 1 or even.

S000442    Number of primes in Collatz sequences having only 1, even numbers, and primes.

S000443    Prime numbers whose Collatz sequence contains only 3 primes.

S000444    Odd numbers n such that evenN(c)/oddN(c) is a new record, where c is the Collatz iteration of n.

S000445    Numbers n such that (5*2^n - 1)/3 is prime.

S000446    Numbers n for which there are no base-2 Fermat pseudoprimes x that have ord(2,x) = n.

S000447*  Irregular table of k! mod p for k = 1..p-1 for primes p.

S000448    Number of distinct values among the p-1 terms k! mod p for k = 1..p-1 for primes p.

S000449    Number of nondistinct values among the p-1 terms k! mod p for k = 1..p-1 for primes p.

S000450    Numbers not in the irregular table of k! mod p for k = 1..p-1 for primes p.

S000451    Least number not in the irregular table of k! mod p for k = 1..p-1 for primes p.

S000452    p - 1 - greatest number not in the irregular table of k! mod p for k = 1..p-1 for primes p.

S000453    Points higher than earlier points in the graph of A229037.

S000454    The x-coordinates of the points in S000453.

S000455    The y-coordinates of the points in S000453.

S000456    Number of ways in which 2n-1 can be written as the sum of 3 odd composite numbers 7 < u < v < w.

S000457    Number of ways in which 2n-1 can be written as the sum of 3 odd composite numbers 9 <= u <= v <= w.

S000458    Number of ways prime(n) is a sum of 3 odd nonprimes r,s,t satisfying 9 <= r <= s <= t.

S000459    Number of ways in which 2n-1 can be written as the sum of 3 odd composite numbers.

S000460    Number of ways in which prime(n) can be written as the sum of 3 odd composite numbers.

S000461    Let s = {1,2,…,n}. Then a(n) = total(d(s))^2 - total(d(s)^3), where d(s) is the number of divisors.

S000462    Same as S000461 except that the set s is a subset of {1,2,…,n}.

S000463    Number of terms in S000461(n) that are zero.

S000464    Sequence S000462 with each row sorted.

S000465    Numbers n such that max(sfp(n), sfp(n+1), sfp(n+2)) = n, where spf(n) is the squarefree part of n.

S000466    Numbers n such that max(sfp(n), sfp(n+1), sfp(n+2)) < sqrt(n), where spf(n) is the squarefree part of n.

S000467*  Numbers n such that max(sfp(n), sfp(n+1), sfp(n+2)) < n^(1/3), where spf(n) is the squarefree part of n.

S000468    Alternating sum of the number of distinct prime factors of the numbers up to n.

S000469    Alternating sum of the number of prime factors (counted multiply) of the numbers up to n.

S000470    Alternating sum of the number of distinct prime factors of the numbers up to 2^n.

S000471    Alternating sum of the number of prime factors (counted multiply) of the numbers up to 2^n.

S000472    Primes p producing the longest possible period in the Fibonacci 3-step (mod p) sequence.

S000473    Primes p producing the longest possible period in the Fibonacci 4-step (mod p) sequence.

S000474    Primes p producing the longest possible period in the Fibonacci 5-step (mod p) sequence.

S000475    Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 1 distinct zero.

S000476    Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 2 distinct zeros.

S000477    Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 1 distinct zero.

S000478    Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 2 distinct zeros.

S000479    Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 3 distinct zeros.

S000480    Record minimum values in A092693 = sum of iterated phi(n).

S000481*  Squares that are the sum of 4 unequal squares of Fibonacci numbers.

S000482    Square root of the numbers in S000481.

S000483    Unique numbers in S000481.

S000484    Square root of the numbers in S000483.

S000485    Five-tuples of squares used in S000481.

S000486    Square root of the five-tuples of squares used in S000481.

S000487    Squares that are the sum of 4 different squares of Fibonacci numbers.

S000488    Square root of numbers in S000487.

S000489    Squares that are the sum of 4 different squares of Fibonacci numbers, with 1 being the least.

S000490    Square root of S000489.

S000491    Squares that are the sum of 4 unequal squares of Lucas numbers.

S000492    Square root of the numbers in S000491.

S000493    Five-tuples of squares used in S000491.

S000494    Square root of the five-tuples used in S000491.

S000495*  Squares that are the sum of 5 unequal squares of Lucas numbers.

S000496    Square root of the numbers in S000495.

S000497    Square root of numbers that are the sum of 5 unequal squares of Lucas numbers, case 1.

S000498    Square root of numbers that are the sum of 5 unequal squares of Lucas numbers, case 2.

S000499    Square root of numbers that are the sum of 5 unequal squares of Lucas numbers, case 3.

S000500    Square root of numbers that are the sum of 5 unequal squares of Lucas numbers, union.

S000501*  Primitive Pythagorean triples in which the two legs are palindromes.

S000502    Primitive Pythagorean triples (in reverse order) in which the two legs are palindromes.

S000503*  Primitive Pythagorean triples in which the hypotenuse and a leg are palindromes.

S000504    Primitive Pythagorean triples (in reverse order) in which the hypotenuse and a leg are palindromes.

S000505*  Primitive Pythagorean triples in which two parts are palindromes.

S000506    Primitive Pythagorean triples (in reverse order) in which two parts are palindromes.

S000507    The hypotenuse of a primitive Pythagorean triangle in which the two legs are palindromes.

S000508    The hypotenuse of a primitive Pythagorean triangle in which the hypotenuse and a leg are palindromes.

S000509    The hypotenuse of primitive Pythagorean triples in which two parts are palindromes.

S000510    Primitive Pythagorean triples in which the hypotenuse and a leg are palindromes in base 2.

S000511    Primitive Pythagorean triples (in reverse order) in which the hypotenuse and a leg are palindromes in base 2.

S000512    The hypotenuse of a primitive Pythagorean triangle in which the hypotenuse and a leg are palindromes in base 2.

S000513    Numerators of a Farey series.

S000514    Denominators of a Farey series.

S000515    The index of the terms in a Farey sequence that are integers.

S000516    The value of the terms in a Farey sequence that are integers.

S000517    Pairs giving the number of primes less than 10^n of the forms 4k+1 and 4k+3.

S000518    Triples giving the number of primes less than 10^n of the forms 6k+1, 6k+3, and 6k+5.

S000519    Quadruples giving the number of primes less than 10^n of the forms 8k+1, 8k+3,…, 8k+7.

S000520    5-tuples giving the number of primes less than 10^n of the forms 10k+1, 10k+3,…, 10k+9.

S000521    6-tuples giving the number of primes less than 10^n of the forms 12k+1, 12k+3,…, 12k+11.

S000522    7-tuples giving the number of primes less than 10^n of the forms 14k+1, 14k+3,…, 14k+13.

S000523    8-tuples giving the number of primes less than 10^n of the forms 16k+1, 16k+3,…, 16k+15.

S000524    16-tuples giving the number of primes less than 10^n of the forms 32k+1, 32k+3,…, 32k+31.

S000525    32-tuples giving the number of primes less than 10^n of the forms 64k+1, 64k+3,…, 64k+63.

S000526*  Firoozbakht conjecture related: ceiling(prime(n)^(1+1/n)) - prime(n+1).

S000527    Conjectured maximum value k such that n = ceiling(prime(k)^(1+1/k)) - prime(k+1), or 0 if none such.

S000528    Conjectured maximum prime p such that n = ceiling(p^(1+1/k)) - nextprime(p), or 0 if none such.

S000529    Number of composite numbers around 2^n.

S000530    Number of composite numbers around 3^n.

S000531    Number of composite numbers around 4^n.

S000532    Number of composite numbers around 5^n.

S000533    Number of composite numbers around 6^n.

S000534    Number of composite numbers around 7^n.

S000535    Number of composite numbers around 8^n.

S000536    Number of composite numbers around 9^n.

S000537    Conjectured sorted differences between adjacent powers.

S000538    Position of S000537(n) in the list of powers A001597.

S000539*  Numbers n such that the distance between Riemann zeros r(n+1) - r(n) is a new minimum.

S000540    Last i for which the difference of the Riemann zeros r(i+1) - r(i) > (7 - n)*I.

S000541*  Irregular triangle of adjacency matrices of simple connected graphs on n points.

S000542    First i for which the difference of the Riemann zeros r(i+1) - r(i) < 2^(-n) * I.

S000543    Number of composite numbers around n!.

S000544    Number of composite numbers around n^n.

S000545    Number of primes in the range n^2 to (n + log(n)/2)^2.

S000546    Number of primes in the range n^2 to (n + 1/log(n))^2.

S000547    Number of primes in the interval p to p + floor(log(p)^2), where p is prime(n).

S000548    Least prime p such that the interval p to p + floor(log(p)^2) contains exactly n primes.

S000549    Number of primitive roots that are minimal.

S000550    The last number having S000549(n) primitive roots.

S000551    Irregular triangle in which row n has the numbers having S000549(n) primitive roots.

S000552    Numbers that occur as the number of primitive roots.

S000553    Sum of the primitive roots of n (mod n).

S000554    The generalized Collatz iteration, n*X+1, is finite for these n, starting from X=3.

S000555    Primes p such that p+2 is prime and p+6 has 2 or fewer prime factors.

S000556    Primes p such that p+2 is prime and p+6 has 3 or fewer prime factors.

S000557    Primes p such that p+2 and p+6 have 2 or fewer prime factors.

S000558    Primes p such that p+2 has 2 or fewer prime factors and p+6 has 3 or fewer prime factors.

S000559    Record pairs (n,k) of odd numbers n that have a record 3x+1 (Collatz) length k.

S000560    The numbers n in S000559.

S000561    The numbers k in S000559.

S000562    The least prime number p such that p+2 is prime and p+6 has n prime factors (counted multiply).

S000563    The least prime number p such that p+2 has up to two prime factors and p+6 has n prime factors.

S000564    Number of composite numbers between S000564(n) and the next prime number.

S000565    Number of composite numbers between S000564(n) and the next prime number.

S000566*  Smallest prime of the form xxxx…xxx n for odd n and digit x, or 0 if there is none.

S000567*  Smallest prime of the form n xxxx…xxx for digit x, or 0 if there is none.

S000568    First prime that begins a range of composite numbers containing n squarefree numbers.

S000569    Number of composite numbers in the range implied by S000568(n).

S000570    First prime that begins a range of composite numbers containing n powers of numbers.

S000571    Composite numbers (p*q*r*…)^k where p, q, r,… are distinct primes and k is any positive integer.

S000572    Triangle of bases for which the least number is palindromic for n bases.

S000573*  Irregular rows of bases for which n is palindromic.

S000574    Primes p for which there is more than one base in which p is palindromic.

S000575    Irregular array of primitive weird numbers of the form 2^n p*q.

S000576    Number of primitive weird numbers of the form 2^n p*q.

S000577*  Least prime p such that the interval p to p + floor(log(p)^2) contains at least n primes.

S000578    Sequence S000577 without duplicates.

S000579    Primitive Pythagorean triples in which the two legs and the hypotenuse are palindromes in base 9, shown in base 10.

S000580*  Primitive Pythagorean triples in which the two legs and the hypotenuse are palindromes in base 9, shown in base 9.

S000581    Number of times n can be written as a square or the sum of a square and a prime.

S000582    The numbers n where records occur in S000581.

S000583    Nonsquare numbers n for which S000581(n) = 1.

S000584    Least k > 1 so that k^n + (k-1)^n >= (k+1)^n + 1^n.

S000585    Pairs of positive numbers x <= y such that x^2 + y^2 is prime.

S000586    Pairs of positive numbers x <= y such that x^4 + y^4 is prime.

S000587    Pairs of positive numbers x <= y such that x^8 + y^8 is prime.

S000588    Pairs of positive numbers x <= y such that x^16 + y^16 is prime.

S000589    Pairs of positive numbers x <= y such that x^32 + y^32 is prime.

S000590    Pairs of positive numbers x <= y such that x^64 + y^64 is prime.

S000591    Pairs of positive numbers x <= y such that x^2 + y^2 is prime and increasing.

S000592    Pairs of positive numbers x <= y such that x^4 + y^4 is prime and increasing.

S000593    Pairs of positive numbers x <= y such that x^8 + y^8 is prime and increasing.

S000594    Pairs of positive numbers x <= y such that x^16 + y^16 is prime and increasing.

S000595    Pairs of positive numbers x <= y such that x^32 + y^32 is prime and increasing.

S000596    Pairs of positive numbers x <= y such that x^64 + y^64 is prime and increasing.

S000597    Number of pairs of numbers (n,x) with 0 < x <= n such that n^2 + x^2 is prime.

S000598    Number of pairs of numbers (n,x) with 0 < x <= n such that n^4 + x^4 is prime.

S000599    Number of pairs of numbers (n,x) with 0 < x <= n such that n^8 + x^8 is prime.

S000600    Number of pairs of numbers (n,x) with 0 < x <= n such that n^16 + x^16 is prime.

S000601    Number of pairs of numbers (n,x) with 0 < x <= n such that n^32 + x^32 is prime.

S000602    Number of pairs of numbers (n,x) with 0 < x <= n such that n^64 + x^64 is prime.

S000603*  Prime numbers that are “popular”.

S000604    The number at which a prime in S000603 first becomes “popular”.

S000605    Irregular table of numbers beginning prime(n)^2 consecutive numbers whose squares sum to square.

S000606    The number of terms in row n of S000605.

S000607    Primes that satisfy p(k+1) - p(k) > p(k+3) - p(k+1), where p(k) is the k-th prime.

S000608    Irregular table of numbers k such that the sum of the squares of n to n+k is a square.

S000609    Irregular table of numbers k such that the sum of the squares of k to n is a square.

S000610    First number having n representations as the sum of up to 9 cubes.

S000611    Conjectured number of numbers requiring exactly n cubes to represent.

S000612*  Conjectured largest number requiring exactly n cubes to represent.

S000613    The product of the composite numbers in the first prime gap having 2*n - 1 numbers.

S000614    Next prime after A000230(n).

S000615    A stable set of primes of the form j^2 + k^2 with all j and k distinct.

S000616    Pairs of numbers (j,k) that produce the primes in S000615.

S000617    A stable set of primes of the form j^4 + k^4 with all j and k distinct.

S000618    Pairs of numbers (j,k) that produce the primes in S000617.

S000619    Number of ways to pair up {1^4, 2^4, ..., (2n)^4 } so the sum of each of the n pairs is prime.

S000620    Irregular table of m! (mod p), where p = prime(n) and m varies from 1 to p.

S000621    Number of numbers <= p that are not among the values m! (mod p), where p = prime(n) and m <= p.

S000622    Primes that produce decreasing values of S000621(k)/prime(k).

S000623    Zero followed by Fibonacci(2*n+1) for n = 1, 2, 3,....

S000624    Least number k such that k^2 + n^2 is a square, or zero if no such square exists.

S000625    Least number k such that k^2 + n^2 is a square and gcd(k,n) = 1, or zero if no such square exists.

S000626    Iterates of the Riesel problem starting at 509203.

S000627*  Numbers that are sums of consecutive squares but not squares.

S000628    Number of numbers between (n-1)^2 and n^2 that are the sum of consecutive squares.

S000629*  Numbers that are sums of consecutive cubes but not cubes.

S000630    Number of numbers between (n-1)^3 and n^3 that are the sum of consecutive cubes.

S000631*  Numbers that are sums of consecutive fourth powers but not fourth powers.

S000632    Number of numbers between (n-1)^4 and n^4 that are the sum of consecutive fourth powers.

S000633*  Numbers that are sums of consecutive fifth powers but not fifth powers.

S000634    Number of numbers between (n-1)^5 and n^5 that are the sum of consecutive fifth powers.

S000635*  Numbers that are sums of consecutive sixth powers but not sixth powers.

S000636    Number of numbers between (n-1)^6 and n^6 that are the sum of consecutive sixth powers.

S000637    The number of ways number n can be written as the sum of a squarefree number and a prime squared.

S000638    Numbers that cannot be written as the sum of a squarefree number and a prime squared.

S000639    Least prime p such that there are n numbers m such that p divides m! + 1.

S000640    Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+3478727)^2 = y^2.

S000641    Sequential product of primes of the form 8k ± 1.

S000642*  Order of the linear difference equation that solves the Diophantine equation x^2 + (x+n)^2 = y^2.

S000643*  Irregular table of difference equations and initial terms descibed in S000642.

S000644    Least lower twin prime of the form 2*n*k ± 1.

S000645    Lower prime of the twin prime pair of the form 16*k ± 1.

S000646    Lower prime of the twin prime pair of the form 24*k ± 1.

S000647    Sorted numbers that are the product of primes of the form 8k ± 1 with non-increasing exponents.

S000648    The order of the difference equation that solves the equation x^2 + (x + S000647(n))^2 = y^2.

S000649    Least k for which the linear difference equation that solves x^2 + (x+k)^2 = y^2 has order 4n-1.

S000650    Largest prime factor of 3^n - 2^n.

S000651    Largest prime factor of 4^n - 3^n.

S000652    Largest prime factor of 5^n - 2^n.

S000653    Largest prime factor of 5^n - 3^n.

S000654    Largest prime factor of 5^n - 4^n.

S000655    Largest prime factor of 6^n - 5^n.

S000656    Largest prime factor of 7^n - 2^n.

S000657    Largest prime factor of 7^n - 3^n.

S000658    Largest prime factor of 7^n - 4^n.

S000659    Largest prime factor of 7^n - 5^n.

S000660    Largest prime factor of 7^n - 6^n.

S000661    Largest prime factor of 8^n - 3^n.

S000662    Largest prime factor of 8^n - 5^n.

S000663    Largest prime factor of 8^n - 7^n.

S000664    Largest prime factor of 9^n - 2^n.

S000665    Largest prime factor of 9^n - 4^n.

S000666    Largest prime factor of 9^n - 5^n.

S000667    Largest prime factor of 9^n - 7^n.

S000668    Largest prime factor of 9^n - 8^n.

S000669    Numbers n such that only one number m exists such that n divides m!+1.

S000670    Primes p such that there are 2 values of m such that p divides m! + 1.

S000671    Primes p such that there are 3 values of m such that p divides m! + 1.

S000672    Primes p such that there are 4 values of m such that p divides m! + 1.

S000673    Primes p such that there are 5 values of m such that p divides m! + 1.

S000674    Primes p such that there are 6 values of m such that p divides m! + 1.

S000675    Primes p such that there are 7 values of m such that p divides m! + 1.

S000676    The composite number A256519(n) divides a(n)! + 1.

S000677    Pairs of numbers A256519(n) and S000676(n).

S000678    The n-th row has the numbers r such that prime(n) divides r! + 1.

S000679    Number of distinct primes among the squares mod prime(n).

S000680    The n-th row has the n primes in the interval 100*k(n) to 100*k(n)+99, where k(n) is in A186311.

S000681*  An encoding of prime constellations.

S000682    Least prime that begins the constellation given by S000681(n).

S000683    Irregular triangle whose n-th row gives the constellation encoded in S000681(n).

S000684    Triangle whose n-th row has n numbers m such that prime S000639(n) divides m! + 1.

S000685    The n+1 degree polynomial which computes A064538(n) times the sum of the n-th powers of positive integers (constant term first).

S000686    The n+1 degree polynomial which computes A064538(n) times the sum of the n-th powers of positive integers (constant term last).

S000687*  Number of times that the sum of 4 unordered triangular numbers equals n.

S000688    Position of first occurrence of n different digits in the number E.

S000689    Position of first occurrence of n different digits in the number Pi.

S000690    Position of the last occurrence of n digits in the number Pi that do not occur earlier.

S000691    Number of integer points in the 3-dimensional sliver in the n x n x n cube.

S000692    Number of integer points in the 4-dimensional sliver in the n x n x n x n 4-cube.

S000693    Number of integer points in the 5-dimensional sliver in the n x n x n x n x n 5-cube.

S000694    Number of divisors of n^2 - 1.

S000695    Sum of the number of divisors of i^2 - 1 for i = 2..n.

S000696    Numbers n at which the number of divisors of n^2 -1 reaches a new record.

S000697    For the numbers n in S000696, the number of divisors of n^2 - 1.

S000698*  Number of palindromic triples that sum to n.

S000699    Number of palindromic pairs that sum to n.

S000700    Number of palindromic (in base 9) triples that sum to n.

S000701    Number of palindromic (in base 8) triples that sum to n.

S000702    Number of palindromic (in base 7) triples that sum to n.

S000703    Number of palindromic (in base 6) triples that sum to n.

S000704    Number of palindromic (in base 5) triples that sum to n.

S000705    Number of palindromic (in base 4) triples that sum to n.

S000706    Number of palindromic (in base 3) triples that sum to n.

S000707    Number of palindromic (in base 2) triples that sum to n.

S000708    Number of palindromic (in base 2) quadruples that sum to n.

S000709*  Lucas entry points: s(n) = smallest m >= 0 such that the n-th prime divides Lucas(m), or -1 if there is no m.

S000710    First n-digit prime in the digits of sqrt(2).

S000711    Position of the first n-digit prime in the digits of sqrt(2).

S000712    First n-digit prime in the digits of sqrt(3).

S000713    Position of the first n-digit prime in the digits of sqrt(3).

S000714    First n-digit prime in the digits of sqrt(Pi).

S000715    Position of the first n-digit prime in the digits of sqrt(Pi).

S000716    Triples of nonnegative (a,b,c) such that a*b-c, b*c-a, and c*a-b are all powers of 2.

S000717    Numbers whose Collatz (3x+1) iteration requires a different number of steps than any smaller number.

S000718    The number of Collatz (3x+1) iterations required for the n-th number in S000717.

S000719    Least prime p such that phi(f(p)) < f(p)/n, where phi is Euler’s totient function and f(p) is the product of the primes up to p.

S000720    Numbers n such that binomial(2n,n) is not divisible by 5, 7, 11, and 13.

S000721    Numbers n such that binomial(2n,n) is not divisible by 7, 11, 13, and 17.

S000722    Numbers n such that binomial(2n,n) is not divisible by 11, 13, 17, and 19.

S000723    Numbers n such that binomial(2n,n) is not divisible by 13, 17, 19, and 23.

S000724    Numbers n such that binomial(2n,n) is not divisible by 17, 19, 23, and 29.

S000725    Largest difference between increasing prime divisors of binomial(2n,n).

S000726    Difference between the the least prime >= n and the greatest prime <= 2*n/3.

S000727    Numbers n such that Lucas(n) is an abundant number.

S000728    Lucas(n) for the numbers n in S000727.

S000729    Numbers n such that Fibonacci(n) and Lucas(n) are abundant numbers.

S000730    GCD of the numbers binomial(2*n, 2*k) for k = 1..n-1.

S000731    GCD of the numbers binomial(2*prime(n), 2*k) for k = 1..prime(n)-1.

S000732    GCD of the numbers binomial(3*n, 3*k) for k = 1..n-1.

S000733    GCD of the numbers binomial(3*prime(n), 3*k) for k = 1..prime(n)-1.

S000734    GCD of the numbers binomial(4*n, 4*k) for k = 1..n-1.

S000735    GCD of the numbers binomial(4*prime(n), 4*k) for k = 1..prime(n)-1.

S000736    GCD of the numbers binomial(5*n, 5*k) for k = 1..n-1.

S000737    GCD of the numbers binomial(5*prime(n), 5*k) for k = 1..prime(n)-1.

S000738    GCD of the numbers binomial(6*n, 6*k) for k = 1..n-1.

S000739    GCD of the numbers binomial(6*prime(n), 6*k) for k = 1..prime(n)-1.

S000740    Primes p such that p = GCD of the numbers binomial(2*p, 2*k) for k = 1..p-1.

S000741    Primes p such that ((3/2)*(p-1)+1) and (3*p-2) are prime.

S000742    Pairs of numbers of the form 2^i * 3^j such that there are no primes between them.

S000743    Triangle of the number of palindromic (in bases 2 to n) triples that sum to n.

S000744    Triangle of the number of palindromic (in bases 2 to n) pairs that sum to n.

S000745    Triangle of the number of palindromic (in bases 2 to n) quadruples that sum to n.

S000746    Prime palindromes that have an increasing number of bases for which they are palindromic.

S000747    Primes p that are not palindromic in any base b with 1 < b < p-1.

S000748    The bases for which S000746(n) is palindromic.

S000749*  Number of times that 2*n can be written as the sum of a prime and a member of P2.

S000750*  Numbers n such that the polynomial x^n - 1 has a divisor of every degree up to n.

S000751    For each n, the highest power in each factor in the factorization of x^n - 1, sorted.

S000752    Primes p for which sum_{1..(p-1)/2} Legendre(i/p) reaches a new maximum.

S000753    For the primes p in S000752, the sum_{1..(p-1)/2} Legendre(i/p), which is a new maximum.

S000754    Primes p congruent to 1 or 5 for which sum_{1..(p-1)/4} Legendre(i/p) reaches a new maximum.

S000755    For the primes p in S000754, the sum_{1..(p-1)/4} Legendre(i/p), which is a new maximum.

S000756    Sequence of pairs (s1,s2) of the distance from a twin prime pair to the next lower and higher prime.

S000757    Prime numbers that are the sum of consecutive triangular numbers.

S000758    Least lower twin prime p such the the sum of the distances to adjacent primes is a new record.

S000759    For the lower twin primes in S000758, the total distance to the nearest primes.

S000760    For the lower twin primes in S000758, the distance to the nearest smaller prime.

S000761    For the lower twin primes in S000758, the distance to the nearest larger prime + 2.

S000762    Primes p for which the polynomial (x^11 - 1)/(x - 1) mod p is irreducible.

S000763    Primes p for which the polynomial (x^13 - 1)/(x - 1) mod p is irreducible.

S000764    Primes p for which the polynomial (x^17 - 1)/(x - 1) mod p is irreducible.

S000765    Primes p for which the polynomial (x^19 - 1)/(x - 1) mod p is irreducible.

S000766    Primes p for which the polynomial (x^23 - 1)/(x - 1) mod p is irreducible.

S000767    Primes p for which the polynomial (x^29 - 1)/(x - 1) mod p is irreducible.

S000768    Primes p for which the polynomial (x^31 - 1)/(x - 1) mod p is irreducible.

S000769    Primes p for which the polynomial (x^37 - 1)/(x - 1) mod p is irreducible.

S000770    Primes p for which the polynomial (x^41 - 1)/(x - 1) mod p is irreducible.

S000771    Primes p for which the polynomial (x^43 - 1)/(x - 1) mod p is irreducible.

S000772    Primes p for which the polynomial (x^47 - 1)/(x - 1) mod p is irreducible.

S000773    Primes p for which the polynomial (x^53 - 1)/(x - 1) mod p is irreducible.

S000774    Primes p for which the polynomial (x^59 - 1)/(x - 1) mod p is irreducible.

S000775    Primes p for which the polynomial (x^61 - 1)/(x - 1) mod p is irreducible.

S000776    Primes p for which the polynomial (x^67 - 1)/(x - 1) mod p is irreducible.

S000777    Primes p for which the polynomial (x^71 - 1)/(x - 1) mod p is irreducible.

S000778    Primes p for which the polynomial (x^73 - 1)/(x - 1) mod p is irreducible.

S000779    Primes p for which the polynomial (x^79 - 1)/(x - 1) mod p is irreducible.

S000780    Primes p for which the polynomial (x^83 - 1)/(x - 1) mod p is irreducible.

S000781    Primes p for which the polynomial (x^89 - 1)/(x - 1) mod p is irreducible.

S000782    Primes p for which the polynomial (x^97 - 1)/(x - 1) mod p is irreducible.

S000783    The first number for which there are n consecutive identical lengths of the Collatz (3X+1) iteration.

S000784    The union of the terms in S000783.

S000785    Irregular table of numbers m such that the sum (k+1)^(2n) + (k+2)^(2n) +…+ (k+m)^(2n) may be prime.

S000786    Prime numbers that are the sum of consecutive 6-th powers.

S000787    Prime numbers that are the sum of consecutive 8-th powers.

S000788    Prime numbers that are the sum of consecutive 10-th powers.

S000789    Prime numbers that are the sum of consecutive 12-th powers.

S000790    Prime numbers that are the sum of consecutive triangular numbers.

S000791    Prime numbers that are the sum of consecutive pentagonal numbers.

S000792    Prime numbers that are the sum of consecutive hexagonal numbers.

S000793    Prime numbers that are the sum of consecutive heptagonal numbers.

S000794    Prime numbers that are the sum of consecutive octagonal numbers.

S000795    Prime numbers that are the sum of consecutive 9-gonal numbers.

S000796    Prime numbers that are the sum of consecutive 10-gonal numbers.

S000797    Prime numbers that are the sum of consecutive 11-gonal numbers.

S000798    Prime numbers that are the sum of consecutive 12-gonal numbers.

S000799    Prime numbers that are the sum of consecutive 13-gonal numbers.

S000800    Record lengths of arithmetic progressions in squarefree numbers beginning with 1.

S000801    Common difference in the arithmetic progression S000800(n).

S000802    Last term in the arithmetic progression listed in S000800(n).

S000803    Triangle whose n-th row has the numbers m such that there is an m-gonal number equal to A063778(n).

S000804    Triangle in which the k-th term in n-th row has index of the S000803(n,k)-gonal number that equals A063778(n).

S000805    Floor(n^(1/4) * sqrt(log(n))).

S000806    Numbers that are not the sum of 3 generalized 7-gonal numbers.

S000807    Numbers that are not the sum of 3 generalized 8-gonal numbers.

S000808    Numbers that are not the sum of 3 generalized 9-gonal numbers.

S000809    Numbers that are not the sum of 3 generalized 10-gonal numbers.

S000810    Numbers that are not the sum of 3 generalized 11-gonal numbers.

S000811    Numbers that are not the sum of 3 generalized 12-gonal numbers.

S000812    Numbers that are not the sum of 3 generalized 13-gonal numbers.

S000813    Numbers that are not the sum of 3 generalized 14-gonal numbers.

S000814    Numbers that are not the sum of 3 generalized 15-gonal numbers.

S000815    Numbers that are not the sum of 3 generalized 16-gonal numbers.

S000816    Numbers that are not the sum of 3 generalized 17-gonal numbers.

S000817    Numbers that are not the sum of 3 generalized 18-gonal numbers.

S000818    Numbers that are not the sum of 3 generalized 19-gonal numbers.

S000819    Numbers that are not the sum of 3 generalized 20-gonal numbers.

S000820    Nondecreasing sequence of numbers whose squares (after the first term) sum to primes.

S000821    Primes formed by sequence S000820.

S000822    All linear second-order sequences are a linear combination of these two sequences.

S000823    All linear third-order sequences are a linear combination of these three sequences.

S000824    All linear fourth-order sequences are a linear combination of these four sequences.

S000825    All linear fifth-order sequences are a linear combination of these five sequences.

S000826    All linear sixth-order sequences are a linear combination of these six sequences.

S000827    All linear seventh-order sequences are a linear combination of these seven sequences.

S000828    All linear eighth-order sequences are a linear combination of these eight sequences.

S000829    All linear ninth-order sequences are a linear combination of these nine sequences.

S000830    All linear tenth-order sequences are a linear combination of these ten sequences.

S000831    All linear 11th-order sequences are a linear combination of these 11 sequences.

S000832*  Sequence of sorted prime quadruples (a,b,c,d) such that a^3 + b^3 + c^3 + d^3 = 2016.

S000833    Sorted edges (pairs of vertices) on a tetrahedron.

S000834    Sorted edges (pairs of vertices a and b with a < b) on a tetrahedron.

S000835    Sorted edges (pairs of vertices) on a cube.

S000836    Sorted edges (pairs of vertices a and b with a < b) on a cube.

S000837    Sorted edges (pairs of vertices) on a octahedron.

S000838    Sorted edges (pairs of vertices a and b with a < b) on an octahedron.

S000839    Sorted edges (pairs of vertices) on a dodecahedron.

S000840    Sorted edges (pairs of vertices a and b with a < b) on a dodecahedron.

S000841    Sorted edges (pairs of vertices) on a icosahedron.

S000842    Sorted edges (pairs of vertices a and b with a < b) on an icosahedron.

S000843    Half of all regular star-polytopes having n sides.

S000844    Regular star-polytopes having n sides.

S000845    Numbers n such that the absolute value of the central term of cyclotomic(n,x) sets a new record.

S000846    Central term in the S000845(n)-th cyclotomic polynomial.

S000847*  Coefficients of the 40755-th cyclotomic polynomial.

S000848    Odd sturdy numbers.

S000849    Even sturdy numbers.

S000850*  Least Hamming weight of a multiple of prime(n).

S000851    Least number k such that k*prime(n) has the least Hamming weight.

S000852    Triangle in which the n-th row is the (n+1)-th prime mod the first n primes.

S000853    The number of primes among the n terms prime(n+1) mod (prime(1)..prime(n)).

S000854    Prime numbers that are not the sum of distinct squares.

S000855    The number n is the first of two consecutive primitive abundant numbers.

S000856    Each row has the n pairs of nonnegative integers whose squares sum to A000446(n).

S000857    Maximum number of primes in a number containing a total of n digits chosen from 1, 3, 7, and 9.

S000858    The least number having n digits (chosen from 1, 3, 7, and 9) containing S000857(n) primes.

S000859    Number of numbers having n digits chosen from 1, 3, 7, and 9 containing S000857(n) primes.

S000860    Least prime dividing NL(n) = sum(k=1..n) 10^(k!-1).

S000861    Least prime dividing RNL(n) = sum(k=1..n) 10^(n!-k!).

S000862    Minimal positive 5-tuple (b1,b2,b3,b4,b5) such that n = b1/b5 + b2/b1 + b3/b2 + b4/b3 + b5/b4.

S000863    Numbers n such that 5 consecutive terms of binomial(n,k) satisfy a cubic polynomial.

S000864    Irregular table containing the distinct prime factors of binomial(2*n,n).

S000865    PrimePi of the irregular table containing the distinct prime factors of binomial(2*n,n).

S000866    Conjectured number of solutions to p^i - q^j = k^2, where p=prime(n), q=prime(n+1), and k an integer.

S000867    a(n) = number of m such that sum of proper divisors of m (A001065(m)) is 2*n+1.

S000868    a(n) = number of m such that sum of proper divisors of m (A001065(m)) is 2*n.

S000869    Mersenne exponents that are part of a twin prime pair.

S000870    Mersenne exponents that are not part of a twin prime pair.

S000871    Where records occur in S000868.

S000872    Where records occur in S000867.

S000873    Difference between binomial(2*n,n) and the closest binomial(m,k) with m > 2*n and 1 < k < n-1.

S000874    Binomial(n,k) with n >= 4 and 2 <= k <= floor(n/2), sorted.

S000875    Powers n such that the sum of the digits in 2^n sets a record.

S000876    Powers n such that the sum of the digits in 3^n sets a record.

S000877    Powers n such that the sum of the digits in 4^n sets a record.

S000878    Powers n such that the sum of the digits in 5^n sets a record.

S000879    Powers n such that the sum of the digits in 6^n sets a record.

S000880    Powers n such that the sum of the digits in 7^n sets a record.

S000881    Powers n such that the sum of the digits in 8^n sets a record.

S000882    Powers n such that the sum of the digits in 9^n sets a record.

S000883*  Number of times n = x^2 + y^2 + z^2 + w^2 with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.

S000884*  Number of times n = x^2 + y^2 + z^2 + w^2 with x + y + z + w a square and with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.

S000885    Records in the number of partitions of n into 4 squares.

S000886    Where records occur in the number of partitions of n into 4 squares.

S000887    Records in the number of partitions of n into 4 squares of integers.

S000888    Where records occur in the number of partitions of n into 4 squares of integers.

S000889    Records in the number of times n = x^2 + y^2 + z^2 + w^2 with x + y + z + w a square and with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.

S000890    Where records occur in the number of times n = x^2 + y^2 + z^2 + w^2 with x + y + z + w a square and with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.

S000891    Pairs (m,k) with increasing m such that m is the first number having k divisors <= sqrt(m).

S000892    The numbers m from sequence S000891.

S000893    The numbers k from sequence S000891.

S000894    Pairs (m,k) with increasing m such that m is the first number having k divisors < sqrt(m).

S000895    The numbers m from sequence S000894.

S000896    The numbers k from sequence S000894.

S000897*  Numbers whose reciprocals have a palindromic repeating part in base 10.

S000898*  Numbers not a multiple of 10 whose reciprocals have a palindromic repeating part.

S000899    Numbers whose reciprocals have a palindromic repeating part in base 6.

S000900    Numbers not a multiple of 6 whose reciprocals have a palindromic repeating part in base 6.

S000901    Numbers whose reciprocals have a palindromic repeating part in base 7.

S000902    Numbers whose reciprocals have a palindromic repeating part in base 8.

S000903    Numbers whose reciprocals have a palindromic repeating part in base 9.

S000904    Least primordial-product number m such that there are n triples (a,b,c) with all a+b+c = k1 and all a*b*c = k2, where k1 and k2 are two constants and k1 is prime.

S000905    The primes occurring in S000904 as the constant k1.

S000906    Primordial-product numbers such that there are no triples (a,b,c) with all a+b+c = k1 and all a*b*c = k2, where k1 and k2 are two positive integers and k1 is prime.

S000907    Numbers n such that 2^n is in S000906.

S000908    For the n-th primordial-product number p, the number of triples (a,b,c) with a+b+c = k1 and a*b*c = p, where k1 is a constant that is the least possible prime (or 0 if there are no primes).

S000909    The primes (or 0) occurring in S000908 as the constant k1.

S000910    Left- or right-truncatable primes in base 9.

S000911    Left- or right-truncatable primes in base 8.

S000912    Left- or right-truncatable primes in base 7.

S000913    Left- or right-truncatable primes in base 6.

S000914    Left- or right-truncatable primes in base 5.

S000915    Left- or right-truncatable primes in base 4.

S000916    Left- or right-truncatable primes in base 3.

S000917    Number of left- or right-truncatable primes in base n.

S000918    Numbers n such that for the Gaussian integer z = n + n*I, sigma(z)/ z is a Gaussian integer.

S000919    Squarefree numbers having at least three prime factors that are in arithmetic progression.

S000920    Squarefree numbers having at least two prime factors that are in arithmetic progression.

S000921    Starting with the complex prime 1+I, the nearest complex prime x + y*I greater than the previous prime.

S000922    The difference between the real and imaginary parts of S000921.

S000923    The squared distance between adjacent points of S000921.

S000924    Squares that are the sum of two repdigit numbers.

S000925    Least number not representable as the sum of n repdigit numbers.

S000926    Least number of repdigit numbers that sum to n.

S000927    Number of ways that number n can be represented as the sum of two nonzero repdigit numbers.

S000928    Number of ways that number n can be represented as the sum of three nonzero repdigit numbers.

S000929    Number of ways that number n can be represented as the sum of four nonzero repdigit numbers.

S000930    Number of ways that number n can be represented as the sum of five nonzero repdigit numbers.

S000931    Number of ways that number n can be represented as the sum of six nonzero repdigit numbers.

S000932    Number of distinct prime factors in the n-th primitive abundant number (A006038).

S000933    Primes of the form x^2 - xy + 2y^2, but not x^2 + xy + 2y^2, with x and y nonnegative.

S000934    In base 3, these positive numbers and their squares are palindromic.

S000935    In base 4, these positive numbers and their squares are palindromic.

S000936    In base 5, these positive numbers and their squares are palindromic.

S000937    In base 6, these positive numbers and their squares are palindromic.

S000938    In base 7, these positive numbers and their squares are palindromic.

S000939    In base 8, these positive numbers and their squares are palindromic.

S000940    In base 9, these positive numbers and their squares are palindromic.

S000941    In base 10, these positive numbers and their squares are palindromic.

S000942    Record pairs (n,k) of numbers n that have a record 3x+1 (Collatz) length k, the number of terms > n.

S000943    Numbers n in S000942.

S000944    Numbers k in S000942.

S000945*  Least twin prime p1 such that there is a smaller twin prime p2 with p1 - p2 = 2*n.

S000946    Least twin prime p2 such that there is a larger twin prime p1 with p1 - p2 = 2*n.

S000947*  First-quadrant complex numbers x + y*I, sorted by norm, that are multiply-perfect.

S000948*  Numbers n such that the complex number n + n*I is a complex multiply-perfect number.

S000949    Norms squared of the complex numbers in S000947.

S000950    Prime factors occurring in the norms given in S000949.

S000951    The odd integers unioned with the odd integers times 2^(2k-1) for k = 1, 2, 3,....

S000952*  Sorted Diophantine quadruples; s(i)*s(j)+1 is a square for each pair s(i) and s(j), i not equal to j.

S000953    The first component of the quadruples in S000952.

S000954    The second component of the quadruples in S000952.

S000955    The third component of the quadruples in S000952.

S000956    The fourth component of the quadruples in S000952.

S000957    Decades in which 4 primes occur.

S000958    Least positive number x that begins a string of n composite numbers of the form x^2 + x + 41.

S000959    Least positive number x that begins a string of exactly n composite numbers of the form x^2 + x + 41.

S000960*  Ways in which a wave can hit an n x n array of points.

S000961    Even numbers which are the sum of two primes of the form x^2 + y^2 + 1.

S000962    Even numbers which are not the sum of two primes of the form x^2 + y^2 + 1.

S000963    Numbers n such that n is in the 3x+1 (Colatz) trajectory of n+1.

S000964    Numbers n such that n-1 is in the 3x+1 (Colatz) trajectory of n.

S000965    Numbers n such that n is in the 3x+1 (Colatz) trajectory of n-1.

S000966    For the Collatz (3x+1) sequence s beginning with n, the difference between n and the next larger number in s, or 0 if there is no larger number.

S000967    For the Collatz (3x+1) sequence s beginning with n, the difference between n and the next smaller number in s.

S000968*  Gaussian integers z = x + i*y, with x > 0 and |y| <= x, whose sum of divisors is z * r, where r is also a Gaussian integer.

S000969    The real and imaginary parts of sigma(z)/z for the complex numbers z in S000968.

S000970    Number of solutions to the Pell equation x^2 - d*y^2 = -1 or 1, where x is tribonacci number t(n).

S000971    Number of solutions to the Pell equation x^2 - d*y^2 = 1, where x is tribonacci number t(n).

S000972    Number of solutions to the Pell equation x^2 - d*y^2 = -1, where x is tribonacci number t(n).

S000973    Irregular table of solutions d to the Pell equation x^2 - d*y^2 = 1, where x is tribonacci number t(n).

S000974    Irregular table of solutions d to the Pell equation x^2 - d*y^2 = -1, where x is tribonacci number t(n).

S000975    Least nonnegative number k such that Lucas(k) begins with the same digits as n.

S000976    Least Lucas number that begins with the same digits as n.

S000977    Odd numbers in the terms of the central hexanomial coefficients.

S000978    Position of odd numbers in the terms of the central hexanomial coefficients.

S000979*  Differences between the position of odd numbers in the terms of the central hexanomial coefficients.

S000980    Primes p such that p^2 divides k^k + (-1)^k (k-1)^(k-1) for some k > 1.

S000981    Least k > 1 such that S000980(n) divides k^k + (-1)^k (k-1)^(k-1).

S000982    Values of k in the range of 1 < k < S000980(n)^2 such that S000980(n)^2 divides k^k + (-1)^k (k-1)^(k-1).

S000983    Differences between terms in the n-th row of S000982.

S000984    Number of terms in each period of S000983.

S000985    Least k > 1 such that prime(n) divides k^k + (-1)^k (k-1)^(k-1).

S000986    Values of k with 1 < k < prime(n)^2 such that prime(n) divides k^k + (-1)^k (k-1)^(k-1).

S000987    Differences between the terms in the n-th row of S000986.

S000988    Number of terms in each period of S000987.

S000989    n^n + (-1)^n (n-1)^(n-1).

S000990    n^n - (-1)^n (n-1)^(n-1).

S000991*  Pairs of numbers x > y > 0 such that x^2 + y^2 is a square, x is as small as possible, and x/y is unique.

S000992    Pairs of numbers x > y > 0 such that x^3 + y^3 is a square, x is as small as possible, and x/y is unique.

S000993    Pairs of numbers x > y > 0 such that x^5 + y^5 is a square, x is as small as possible, and x/y is unique.

S000994    Pairs of numbers x > y > 0 such that x^7 + y^7 is a square, x is as small as possible, and x/y is unique.

S000995    Pairs of numbers x > y > 0 such that x^9 + y^9 is a square, x is as small as possible, and x/y is unique.

S000996*  Triples x >= y >= z that form the sides of a primitive triangle having integer area.

S000997    Triples x <= y <= z that form the sides of a primitive triangle having integer area.

S000998    Areas of the triangles given in S000996.

S000999    Integer areas of the primitive triangles having integer sides.

S001000    Triples (x,y,z) with 1 < x < y < z such that x! y! z! is a square.

S001001    Triples (x,y,z) with 1 < x < y < z and z = y + 3 such that x! y! z! is a square.

S001002    Triples (x,y,z) with 1 < x < y < z and z = y + 2 such that x! y! z! is a square.

S001003    Triples (x,y,z) with 1 < x < y < z and z = y + 1 such that x! y! z! is a square.

S001004*  Least triple of palindromes x <= y <= z such that n = x + y + z.

S001005    The first number in the triple S001004(n).

S001006    Numbers n such that S001005(n) > 0.

S001007    Numbers x such that there is a primitive triangle whose sides are x, x-1, and 3.

S001008    Numbers x such that there is a primitive triangle whose sides are x, x-2, and 4.

S001009    Numbers x such that there is a primitive triangle whose sides are x, x-1, and 5.

S001010    Numbers x such that there is a primitive* triangle whose sides are x, x-2, and 6.

S001011    Numbers x such that there is a primitive triangle whose sides are x, x-1, and 7.

S001012    Numbers x such that there is a primitive triangle whose sides are x, x-2, and 8.

S001013    Numbers x such that there is a primitive triangle whose sides are x, x-1, and 9.

S001014    Numbers x such that there is a primitive* triangle whose sides are x, x-2, and 10.

S001015    Palindromes that are either non-decreasing or non-increasing toward their center.

S001016    Record Collatz (3x+1) pairs (see comments).

S001017    Numbers c from the sequence S001016.

S001018    Number of terms greater than c in the 3x+1 iteration starting with c = S001017(n).

S001019    For odd n, the number of Collatz (3x+1) iteration terms greater than n.

S001020    For even n, the number of Collatz (3x+1) iteration terms greater than n.

S001021    Smaller twin prime p such that phi(p-1) > phi(p+1), where phi is Euler’s totient function.

S001022    Smaller twin prime p such that phi(p-1) < phi(p+1), where phi is Euler’s totient function.

S001023    Smaller twin prime p such that phi(p-1) = phi(p+1), where phi is Euler’s totient function.

S001024    Smaller twin prime p such that phi(p-1) >= phi(p+1), where phi is Euler’s totient function.

S001025    Smaller twin prime p such that phi(p-1) <= phi(p+1), where phi is Euler’s totient function.

S001026    Triples 0 < a <= b < c with a + b = c, gcd(a,b) = 1, and log(c)/log(rad(a*b*c)) a new maximum.

S001027    Numbers that are the sum of cubes of four primes.

S001028    Least number that is the sum of four cubes of primes in n ways.

S001029    The n 4-tuples of primes whose cubes sum to S001028(n).

S001030    Numbers not represented by the sum of 7 positive cubes.

S001031    Prime numbers whose reverse in base 10 is also prime.

S001032    Prime numbers containing only digits 0 and 1 whose reverse in base 10 is also prime.

S001033*  Numbers that are not the sum of 17 nonnegative 5-th powers.

S001034    Numbers that are not the sum of 16 nonnegative 5-th powers.

S001035    Numbers that are not the sum of 15 nonnegative 5-th powers.

S001036    Numbers that are not the sum of 14 nonnegative 5-th powers.

S001037    Numbers that are not the sum of 13 nonnegative 5-th powers.

S001038    Numbers that are not the sum of 12 nonnegative 5-th powers.

S001039    Numbers that are not the sum of 11 nonnegative 5-th powers.

S001040    Numbers that are not the sum of 10 nonnegative 5-th powers.

S001041    Numbers that are not the sum of 9 nonnegative 5-th powers.

S001042    Number of numbers between (n-1)^5 and n^5 that are not the sum of 8 nonnegative fifth powers.

S001043    Numbers that are not the sum of 8 nonnegative fifth powers.

S001044    The last 1000 numbers that are not the sum of 8 nonnegative fifth powers.

S001045    Numbers that are not the sum of 18 nonnegative 5-th powers.

S001046    Numbers that are not the sum of 19 nonnegative 5-th powers.

S001047    Numbers that are not the sum of 20 nonnegative 5-th powers.

S001048    Numbers that are not the sum of 21 nonnegative 5-th powers.

S001049    Numbers that are not the sum of 22 nonnegative 5-th powers.

S001050    Numbers that are not the sum of 23 nonnegative 5-th powers.

S001051    Numbers that are not the sum of 24 nonnegative 5-th powers.

S001052    Numbers that are not the sum of 25 nonnegative 5-th powers.

S001053    Numbers that are not the sum of 26 nonnegative 5-th powers.

S001054    Numbers that are not the sum of 27 nonnegative 5-th powers.

S001055    Number of numbers between (n-1)^5 and n^5 that are not the sum of 7 nonnegative fifth powers.

S001056    Numbers that are not the sum of 7 nonnegative fifth powers.

S001057    Last number requiring 37-n positive fifth powers to represent.

S001058    Irregular triangle of numbers requiring 37-n positive fifth powers to represent.

S001059    Number of numbers between (n-1)^5 and n^5 that are not the sum of 6 nonnegative fifth powers.

S001060    Numbers that are not the sum of 6 nonnegative fifth powers.

S001061    Primitive Pythagorean quadruples (d, c, b, a) sorted so that a <= b <= c < d.

S001062    Primitive Pythagorean quadruples (a, b, c, d) sorted so that a <= b <= c < d.

S001063    Primitive Pythagorean quintuples (a, b, c, d, e) sorted so that a <= b <= c <= d < e.

S001064    Primitive Pythagorean sextuples (a, b, c, d, e, f) sorted.

S001065    Primitive Pythagorean septuples (a, b, c, d, e, f, g) sorted.

S001066    Primitive Pythagorean octuples (a, b, c, d, e, f, g, h) sorted.

S001067    Primitive Pythagorean 9-tuples (a, b, c, d, e, f, g, h, i) sorted.

S001068    Primitive Pythagorean 10-tuples (a, b, c, d, e, f, g, h, i, j) sorted.

S001069    Primitive Pythagorean 11-tuples (a, b, c, d, e, f, g, h, i, j, k) sorted.

S001070    Numbers n such that the digits of prime(1) to prime(n) concatenated in some order is possibly a palindrome.

S001071    Numbers n such that the digits of 1 to n concatenated in some order is possibly a palindrome.

S001072    The last 1000 numbers that are not the sum of fourth powers of distinct primes.

S001073    Number of numbers between prime(n)^4 and prime(n+1)^4 that are not the sum of distinct fourth powers of primes.

S001074    Smallest positive number s(n) such that s(n) and all greater numbers can be expressed as the sum of n-th powers of distinct primes.

S001075    For each odd prime p, the p least-multiples of p whose prime factors are in arithmetic progression.

S001076    For each odd prime p, the p common differences of the arithmetic progressions mentioned in S001075.

S001077    Numbers n for which a record number of numbers binomial(n,m) are squarefree, where m <= n/2.

S001078    The number of numbers for which binomial(n,m) is squarefree, where n is in S001077 and m <= n/2.

S001079    Irregular triangle of numbers m for which binomial(S001077(n),m) is squarefree.

S001080    Triangular table in which the n-th row has the n numbers that are the larger of a pair of numbers x and y such that x^2 - y^2 = A094191(n).

S001081    Triangular table in which the n-th row has the n numbers that are the smaller of a pair of numbers x and y such that x^2 - y^2 = A094191(n).

S001082    Numbers n such that binomial(2n,n) is fourth-power free.

S001083    Numbers n such that binomial(2n,n) is fifth-power free.

S001084*  Pairs of numbers (n,k) such that there is no n-gonal k-gonal number greater than 1, sorted by the sum n+k and then n.

If any of these sequences are used in some manner, the page describing the sequence must be referenced. For example:

    T. D. Noe, Sequence S000687, Integer Sequences, (www.IntegerSequences.org/s000687.html). 

For the most part, the sequences were not part of the OEIS database (www.oeis.org) when they were added to this website. Contact: noe (at) sspectra.com

* nice sequences