cp's OEIS Frontend

If k is a positive term then k is even (or else k has no generator in base k+1) but not a multiple of 4 (or else k has no generator in base k/2). - David Applegate, Jan 09 2022. See A349821 and A350607 for the k/2 and (k-2)/4 sequences.

It is not known if this sequence is infinite.

The eight terms 10 through 206 are all twice primes (cf. A349820).

It is not known if A230624 is infinite. Many of its initial terms are twice primes, so it would interesting if these primes could be characterized in some other way.

a(n)+1 typically has a slowly growing power of 2 as factor. as can be seen here: (PARI) for(k=1, #a, print1(valuation(a[k]+1,2),", ")): 1, 3, 2, 2, 5, 4, 4, 3, 6, 4, 7, 5, 5, 5, 7, 6, 6, 6, 6, 7, 10, 9, 8, 9, 8, 8, 9, 9, 12, 11, 11, 10, 10, 11, 10, 17, 11, 11, 12, 12,.. - Hugo Pfoertner Jan 03 2022

This sequence could certainly be divided by 2, just as we divided A230624 itself by 2 to get A349821. But there is a reason for not dividing this by 2: it appears that, for any power of 2, from a certain point on, the sequence is divisible by that power of 2. At present this only a conjecture. But it may provide a clue to the structure of this sequence and therefore of A230624.

For example, after 14 terms, the present sequence (as far as it is presently known) can be divided by 64, giving 3, 1, 8, 2, 1, 1, 2, 1, 4, 6, 1, 5, 4, 5, 6, 12, 6, 4, 6, 2, 14, 32, 8, 2, 2, 20, 32, 4, 4, 20, 62, 60, 36, 284, 48, 424, 440, 40, 136, 16, 200, 48, 96, 40, ..., which in turn can be divided by 2 after a further 14 terms.

So there is at least some structure here.