Numbers n such that binomial(2n,n) is not divisible by 13, 17, 19, and 23.
0, 1, 2, 3, 4, 5, 6, 57, 149754111, 149754112, 149754113, 149754127, 149754128, 149754129, 149754130, 149755193, 149755194, 149755195, 149755196, 149787969, 151179966, 151179967, 151179968, 151179969, 151180321, 151180341, 151180342, 151180343, 151180344
1
The sequence for the 3-5-7-11 case has only 3 terms. These related five sequences (S000720-S000724) provide more insight to the problem. Using the analysis of the Pomerance paper, this sequence is expected to be infinite.
T. D. Noe, Plot of 133 terms
T. D. Noe, Table of 133 terms
Carl Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 122 (2015), pp. 636-644.
(Mma) lim = 1000000; Intersection[Table[FromDigits[IntegerDigits[k, 7], 13], {k, 0, lim}], Table[FromDigits[IntegerDigits[k, 9], 17], {k, 0, lim}], Table[FromDigits[IntegerDigits[k, 10], 19], {k, 0, lim}], Table[FromDigits[IntegerDigits[k, 12], 23], {k, 0, lim}]]
Cf. A151750 (the 3,5,7,11 case), S000720-S000724.
nonn
T. D. Noe, Oct 16 2015