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A363594 a(n) = the n-th instance of b(k)/2 such that b(k-1) and b(k-2) are both odd, where b(n) = A359804(n).

%I A363594 #11 Jul 06 2023 20:56:29
%S A363594 2,4,8,13,16,17,19,23,26,29,31,32,34,37,38,41,43,46,47,53,58,59,61,62,
%T A363594 64,67,68,71,73,74,76,79,82,83,86,89,92,94,97,101,103,106,107,109,113,
%U A363594 116,118,122,124,127,128,131,134,136,137,139,142,146,148,149,151,152,157,158,163,164,166,167,172
%N A363594 a(n) = the n-th instance of b(k)/2 such that b(k-1) and b(k-2) are both odd, where b(n) = A359804(n).
%C A363594 The sequence strictly increases, a consequence of definition of A359804.
%C A363594 Conjecture: { A000079 \ {1} } U { A000040 \ {3, 5, 7, 11} } is a subset. In other words, this sequence is the union of powers of 2 greater than 1, and primes greater than 11.
%C A363594 This sequence is conjectured to be infinite. It tracks all occurrences of consecutive odd terms in A359804, which are (by definition) always followed by an even term, from which a(n) is derived. - _David James Sycamore_, Jun 21 2023
%H A363594 Michael De Vlieger, <a href="/A363594/b363594.txt">Table of n, a(n) for n = 1..10000</a>
%F A363594 a(n) = A359804(A363593(n)+2)/2 = A361639(A363593(n)+1).
%e A363594 a(1) = 2 since b(3..5) = {3, 5, 4}; 4/2 = 2.
%e A363594 a(2) = 4 since b(8..10) = {7, 9, 8}; 8/2 = 4.
%e A363594 a(3) = 8 since b(22..24) = {33, 35, 16}; 16/2 = 8.
%e A363594 a(4) = 13 since b(29..31) = {45, 49, 26}; 26/2 = 13.
%e A363594 a(5) = 16 since b(36..38) = {55, 63, 32}; 32/2 = 16, etc.
%t A363594 nn = 500; c[_] = False; q[_] = 1;
%t A363594 Set[{i, j}, {1, 2}]; c[1] = c[2] = True; q[2] = 2; u = 3;
%t A363594 Reap[Do[
%t A363594     (k = q[#]; While[c[k #], k++]; k *= #;
%t A363594        While[c[# q[#]], q[#]++]) &[(p = 2;
%t A363594       While[Divisible[i j, p], p = NextPrime[p]]; p)];
%t A363594     If[OddQ[i j], Sow[k/2]];
%t A363594     Set[{c[k], i, j}, {True, j, k}];
%t A363594     If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, -1]]
%Y A363594 Cf. A000040, A000079, A100484, A359804, A361639, A363593.
%K A363594 nonn
%O A363594 1,1
%A A363594 _Michael De Vlieger_ and _David James Sycamore_, Jun 12 2023