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A386314 a(1) = 1 and thereafter a(n) is the smallest number k of the form 6*x+-1 not already in the sequence but where the reduced Collatz step A139391(k) is in the sequence.

%I A386314 #65 Aug 25 2025 11:34:06
%S A386314 1,5,13,17,11,7,29,19,25,37,49,53,35,23,61,65,43,77,85,101,67,89,59,
%T A386314 113,133,149,157,173,115,181,197,131,205,209,139,185,229,241,245,163,
%U A386314 217,269,179,119,79,277,289,301,305,203,317,211,281,187,325,341,227,151,349,373
%N A386314 a(1) = 1 and thereafter a(n) is the smallest number k of the form 6*x+-1 not already in the sequence but where the reduced Collatz step A139391(k) is in the sequence.
%C A386314 These numbers are the Collatz pre-images in the form 6*x +- 1 of all previous terms not already in the sequence.
%C A386314 The pre-images of a term t are all p which reach t by a single odd to odd step A139391(p) = t.
%C A386314 These pre-images are those p = (t*2^k-1)/3 with k>=0 which are odd integers, and with here t != 0 (mod 3) there are infinitely many p != 0 (mod 3) for each t.
%C A386314 Multiples of 3 have no odd pre-images and are excluded here in order to have the essential part of the tree of odd to odd descents.
%C A386314 The trajectory of a term t reaches 1 by steps to successively earlier terms in this sequence (at various distances apart).
%C A386314 If the Collatz conjecture is true, then this sequence is permutation of the numbers of the form 6x +- 1 (A007310).
%H A386314 David A. Corneth, <a href="/A386314/b386314.txt">Table of n, a(n) for n = 1..10000</a> (using Michel Marcus' code)
%H A386314 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a386/A386314.java">Java program</a> (github)
%e A386314 a(3) = 13, since 13 (a pre-image of a(2) = 5) is the smallest unused pre-image of a(1) and a(2).
%e A386314 a(10) = 37 since 37 (a pre-image of a(6) = 7) is the smallest unused pre-image of all previous terms.
%o A386314 (PARI) lista(nn) = my(va=List(1), vs = Map(), imin=1, i=imin, nb=1); mapput(vs, 1, 1); while(#va<nn, my(k=3*i+1+i%2, y=3*k+1); y=y/2^valuation(y, 2); if(mapisdefined(vs, y) && !mapisdefined(vs, k), listput(va, k); nb++; mapput(vs, k, nb); if(i==imin, imin++); i=imin, i++;);); va; \\ _Michel Marcus_, Aug 25 2025
%Y A386314 Cf. A007310, A178414, A178415, A139391.
%K A386314 nonn,new
%O A386314 1,2
%A A386314 _Jules Beauchamp_, Jul 18 2025