cp's OEIS Frontend

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.

This is a subsequence of A002144, A002313, A003655, A050150, A062090, A141293, A175768, A192592, A226181 (conjectural).

All terms are odd. If even n belongs to this sequence, then n-1 is odd and thus (n-1)/A007733(n) is also odd and thus must be equal to 1. On the other hand, for even n, A007733(n) < n/2 <= n-1, i.e., (n-1)/A007733(n) > 1, a contradiction.

Subsequence of A001567.

Contains all composite Fermat numbers A000215(k) = 2^(2^k)+1 (which are composite for 5<=k<=32 and conjecturally for any k>=5). In particular, a(5) = A000215(5), a(12) = A000215(6), and a(13) <= A000215(7) = 2^128+1.

Pseudoprimes n such that (n-1)/ord_{n}(2) = 2^k for some k, where ord_{n}(2) = A002326((n-1)/2) is the multiplicative order of 2 mod n. Composite numbers n such that Od(ord_{n}(2)) = Od(n-1), where ord_{n}(2) as above and Od(m) = A000265(m) is the odd part of m. Note that if Od(ord_{n}(2)) = Od(n-1), then ord_{n}(2)|(n-1). - Thomas Ordowski, Mar 13 2019