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

1,1

Comments

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

Links

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

Formula

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

Example

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.

Mathematica

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]]

Crossrefs

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

Sequence in context: A133802 A076202 A268713 * A320630 A224399 A114398

Adjacent sequences: A363591 A363592 A363593 * A363595 A363596 A363597

Keyword

nonn

Author

Michael De Vlieger and David James Sycamore, Jun 12 2023

Status

approved