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

Author

Joseph L. Pe, Dec 29 2002

Keywords

• nonn

Comments

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

Examples

			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.
		

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

Crossrefs

Cf. A006370, A006577.

Programs

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 *)