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A384327 Minimal Trips Around The Collatz Galaxy: a(n) is the minimal cycle length containing n. Each step in the cycle must be either to the next larger integer or follow a Collatz trajectory: k -> 3k+1 if k is odd or k -> k/2 if k is even.

%I A384327 #23 Jun 24 2025 00:51:24
%S A384327 2,2,3,3,4,4,5,4,5,4,5,5,6,5,6,4,6,6,7,6,7,5,6,6,8,7,8,5,7,7,8,8,9,6,
%T A384327 7,7,9,9,10,6,8,8,9,9,11,7,8,8,10,10,11,7,9,9,10,10,12,8,9,9,11,11,12,
%U A384327 8,10,10,11,11,13,9,10,10,12,12,13,9,11,11,12
%N A384327 Minimal Trips Around The Collatz Galaxy: a(n) is the minimal cycle length containing n. Each step in the cycle must be either to the next larger integer or follow a Collatz trajectory: k -> 3k+1 if k is odd or k -> k/2 if k is even.
%C A384327 Excellent puzzle to get elementary students engaged in arithmetic. Recommended for use after the Collatz conjecture (3x+1 conjecture) is introduced side-by-side with its 3x-1 cousin.
%C A384327 In a tree of possible choices from n every positive integer occurs at most once which allows for lots of cutting in the search space. - _David A. Corneth_, May 26 2025
%H A384327 David A. Corneth, <a href="/A384327/b384327.txt">Table of n, a(n) for n = 1..10000</a>
%H A384327 David A. Corneth, <a href="/A384327/a384327.gp.txt">list of n, a(n) and the (or a) shortest path of length a(n)</a>
%H A384327 David A. Corneth, <a href="/A384327/a384327_1.gp.txt">PARI program</a>
%H A384327 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A384327 a(25) is 8 because the shortest cycle containing 25 has 8 elements: 25 => 26 => 13 => 14 => 15 => 46 => 23 => 24 => 25.
%o A384327 (PARI) \\ See Corneth link
%Y A384327 Cf. A006370.
%K A384327 nonn
%O A384327 1,1
%A A384327 _Gordon Hamilton_, May 26 2025
%E A384327 More terms from _Alois P. Heinz_, May 26 2025