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

1,1

Comments

Recall that f(n) = n/2 if n is even; = 3n + 1 if n is odd.

Links

Table of n, a(n) for n=1..39.

Example

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.

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]&]

Crossrefs

Cf. A006577, A078419, A078420.

Sequence in context: A379436 A240049 A387767 * A100300 A027296 A179601

Adjacent sequences: A078415 A078416 A078417 * A078419 A078420 A078421

Keyword

nonn

Author

Joseph L. Pe, Dec 29 2002

Extensions

Extended by Robert G. Wilson v, Dec 30 2002

Name clarified by Sean A. Irvine, Jun 29 2025

Status

approved