The first number for which there are n consecutive identical lengths of the Collatz (3X+1) iteration.
1, 12, 28, 98, 98, 386, 943, 1494, 1680, 2987, 2987, 2987, 2987, 2987, 7083, 7083, 7083, 57346, 57346, 57346, 57346, 57346, 57346, 57346, 57346, 252548, 252548, 331778, 331778, 524289, 596310, 596310, 596310, 596310, 596310, 596310, 596310, 596310, 596310, 596310
1
That is, the lengths of the iterations Collatz(s(n)), Collatz(s(n)+1),…, Collatz(s(n)+n-1) are all the same.
T. D. Noe, Plot of 176 terms
T. D. Noe, Table of 176 terms
Wikipedia, Collatz conjecture
(Mma) Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t2 = {}; mx = 0; last = 1; cnt = 1; Do[c = Length[Collatz[n]]; If[c == last, cnt++, If[cnt > mx, mx = cnt; AppendTo[t2, {n - cnt, cnt}]]; last = c; cnt = 1], {n, 2, 100000}]; last = 0; Flatten[Table[s = Table[p[[1]], {p[[2]] - last}]; last = p[[2]]; s, {p, t2}]]
Cf. S000784.
nonn
T. D. Noe, Dec 02 2015