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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.)

%I A078417 #31 Feb 16 2025 08:32:48
%S A078417 12,14,18,20,22,28,29,34,36,37,44,45,49,50,52,54,60,62,65,66,68,69,76,
%T A078417 78,82,84,86,92,94,98,99,100,101,108,109,114,116,118,124,125,130,131,
%U A078417 132,133,140,142,145,146,148,150,156,157,162,164,165,172,173,177,178
%N 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.)
%C A078417 Recall that f(k) = k/2 if k is even, 3k + 1 if k is odd (A006370).
%H A078417 Eric M. Schmidt, <a href="/A078417/b078417.txt">Table of n, a(n) for n = 1..10000</a>
%H A078417 Marcus Elia and Amanda Tucker, <a href="http://arxiv.org/abs/1511.09141">Consecutive Integers and the Collatz Conjecture</a>, arXiv:1511.09141 [math.NT], 2015.
%H A078417 Lynn E. Garner, <a href="http://dx.doi.org/10.1016/S0012-365X(85)80020-0">On heights in the Collatz 3n+1 problem</a>, Discrete Math, 55 (1985), 57-64.
%H A078417 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H A078417 Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz Conjecture</a>
%H A078417 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A078417 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.
%p A078417 collatz:= proc(n) option remember; `if`(n=1, 0,
%p A078417    1 + collatz(`if`(n::even, n/2, 3*n+1)))
%p A078417 end:
%p A078417 q:= n-> is(collatz(n)=collatz(n+1)):
%p A078417 select(q, [$1..200])[];  # _Alois P. Heinz_, Jul 19 2023
%t A078417 h[n_] := Length@NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &];
%t A078417 okQ[n_] := h[n] == h[n+1];
%t A078417 Select[Range[200], okQ] (* _Jean-François Alcover_, Jan 12 2024 *)
%Y A078417 Cf. A006370, A006577.
%K A078417 nonn
%O A078417 1,1
%A A078417 _Joseph L. Pe_, Dec 29 2002