A078418 Numbers k such that h(k) = h(k-1) + h(k-2), where h(k) = A006577(k) + 1 is the length of the sequence {k, f(k), f(f(k)), ...., 1} in the Collatz (or 3x + 1) problem. (The earliest "1" is meant.)
6, 22, 97, 108, 114, 495, 559, 2972, 3092, 3124, 3147, 3154, 3329, 3367, 3483, 3643, 3711, 3748, 3756, 3982, 4009, 4767, 17435, 17782, 17796, 17863, 17892, 17897, 18079, 18139, 18422, 18580, 18644, 18688, 18784, 18804, 18952, 19739, 19868
Offset: 1
Keywords
Examples
n, f(n), f(f(n)), ...., 1 for n = 22, 21, 20, respectively, are: 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1; 21, 64, 32, 16, 8, 4, 2, 1; 20, 10, 5, 16, 8, 4, 2, 1. Hence h(22) = 16 = 8 + 8 = h(21) + h(20) and 22 belongs to the sequence.
Programs
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Mathematica
f[n_] := If[EvenQ[n], n/2, 3n+1]; h[n_] := Module[{a, i}, i=n; a=1; While[i>1, a++; i=f[i]]; a]; Select[Range[3, 19900], h[ # ]==h[ #-1]+h[ #-2]&]
Extensions
Extended by Robert G. Wilson v, Dec 30 2002
Name clarified by Sean A. Irvine, Jun 29 2025
Comments