A363593 -id:A363593 - cp's OEIS frontend

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).

2, 4, 8, 13, 16, 17, 19, 23, 26, 29, 31, 32, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 92, 94, 97, 101, 103, 106, 107, 109, 113, 116, 118, 122, 124, 127, 128, 131, 134, 136, 137, 139, 142, 146, 148, 149, 151, 152, 157, 158, 163, 164, 166, 167, 172

Offset: 1

Michael De Vlieger and David James Sycamore, Jun 12 2023

nonn

The sequence strictly increases, a consequence of definition of A359804.
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.
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

			a(1) = 2 since b(3..5) = {3, 5, 4}; 4/2 = 2.
a(2) = 4 since b(8..10) = {7, 9, 8}; 8/2 = 4.
a(3) = 8 since b(22..24) = {33, 35, 16}; 16/2 = 8.
a(4) = 13 since b(29..31) = {45, 49, 26}; 26/2 = 13.
a(5) = 16 since b(36..38) = {55, 63, 32}; 32/2 = 16, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A000079, A100484, A359804, A361639, A363593.

nn = 500; c[] = False; q[] = 1;
Set[{i, j}, {1, 2}]; c[1] = c[2] = True; q[2] = 2; u = 3;
Reap[Do[
    (k = q[#]; While[c[k #], k++]; k *= #;
       While[c[# q[#]], q[#]++]) &[(p = 2;
      While[Divisible[i j, p], p = NextPrime[p]]; p)];
    If[OddQ[i j], Sow[k/2]];
    Set[{c[k], i, j}, {True, j, k}];
    If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, -1]]

a(n) = A359804(A363593(n)+2)/2 = A361639(A363593(n)+1).

cp's OEIS Frontend

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).

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