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A261690 a(1) = 1; for n>1, a(n) is the smallest number not already present which is entailed by the rules (i) k present => 3*k+1 present; (ii) 2*k present => k present.

%I A261690 #39 Sep 03 2015 10:45:06
%S A261690 1,4,2,7,13,22,11,34,17,40,20,10,5,16,8,25,31,49,52,26,61,67,76,38,19,
%T A261690 58,29,79,88,44,94,47,103,115,121,133,142,71,148,74,37,112,56,28,14,
%U A261690 43,85,130,65,157,169,175,184,92,46,23,70,35,106,53,139,160,80
%N A261690 a(1) = 1; for n>1, a(n) is the smallest number not already present which is entailed by the rules (i) k present => 3*k+1 present; (ii) 2*k present => k present.
%C A261690 An analog of A109732 such that the statement 'the sequence is a permutation of the positive integers not divisible by 3' is equivalent to the (3*n+1)-conjecture for numbers not divisible by 3.
%C A261690 On Aug 29 2015, _Max Alekseyev_ noted that, while the (3*n+1)-conjecture indeed implies that the sequence is a permutation of the positive integers not divisible by 3, the opposite statement is an open question. The author cannot yet prove this, so his previous comment is only a conjecture.
%C A261690 In connection with this, consider the following conjecture which could be called the (n-1)/3-conjecture. Let n be any number not divisible by 3. If n==1 (mod 3) and (n-1)/3 is not divisible by 3, then set n_1 = (n-1)/3. Otherwise set n_1 = 2*n. Conjecture. There exists an iteration n_m = 1. Does the (n-1)/3-conjecture imply the (3*n+1)-conjecture?
%C A261690 Example: 19->38->76->25->8->16->5->10->20->40->13->4->1.
%H A261690 Peter J. C. Moses, <a href="/A261690/b261690.txt">Table of n, a(n) for n = 1..10000</a>
%Y A261690 Cf. A006577, A006877, A006884, A109732.
%K A261690 nonn
%O A261690 1,2
%A A261690 _Vladimir Shevelev_, Aug 28 2015