A078417 Numbers k such that h(k) = h(k+1), where h(k) is the length of k, f(k), f(f(k)), ..., 1 in the Collatz (or 3x + 1) problem. (The earliest "1" is meant.)

12, 14, 18, 20, 22, 28, 29, 34, 36, 37, 44, 45, 49, 50, 52, 54, 60, 62, 65, 66, 68, 69, 76, 78, 82, 84, 86, 92, 94, 98, 99, 100, 101, 108, 109, 114, 116, 118, 124, 125, 130, 131, 132, 133, 140, 142, 145, 146, 148, 150, 156, 157, 162, 164, 165, 172, 173, 177, 178

Offset

1,1

Comments

Recall that f(k) = k/2 if k is even, 3k + 1 if k is odd (A006370).

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Marcus Elia and Amanda Tucker, Consecutive Integers and the Collatz Conjecture, arXiv:1511.09141 [math.NT], 2015.

Lynn E. Garner, On heights in the Collatz 3n+1 problem, Discrete Math, 55 (1985), 57-64.

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz Conjecture

Index entries for sequences related to 3x+1 (or Collatz) problem

Example

The Collatz trajectories k, f(k), f(f(k)), ..., 1 for k = 12 and 13, respectively, are {12, 6, 3, 10, 5, 16, 8, 4, 2, 1} and {13, 40, 20, 10, 5, 16, 8, 4, 2, 1}, which are both of length 10. Hence h(12) = h(13) = 10, so 12 belongs to this sequence.

Maple

collatz:= proc(n) option remember; `if`(n=1, 0,
   1 + collatz(`if`(n::even, n/2, 3*n+1)))
end:
q:= n-> is(collatz(n)=collatz(n+1)):
select(q, [$1..200])[];  # Alois P. Heinz, Jul 19 2023

Mathematica

h[n_] := Length@NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &];
okQ[n_] := h[n] == h[n+1];
Select[Range[200], okQ] (* Jean-François Alcover, Jan 12 2024 *)

Crossrefs

Cf. A006370, A006577.

Sequence in context: A208156 A135770 A091989 * A107835 A257966 A342814

Adjacent sequences: A078414 A078415 A078416 * A078418 A078419 A078420

Keyword

nonn

Author

Joseph L. Pe, Dec 29 2002

Status

approved