[seqfan] Primitive sequences

[Charles Greathouse]

When I find a set which is closed under (a) multiplication by terms of that
set or (b) multiplication by any positive integer I ask myself about the
primitive terms of the set, those which are not (a) the multiple of two
smaller elements or (b) not the product of a smaller element and a positive
integer.

Yesterday I asked myself that question about A035251, positive numbers of
the form x^2 - 2y^2. The primitive terms (in the sense of a) seem to be
A055673, which can be characterized as
Absolute values of norms of primes in ring of integers Z[sqrt(2)].
or as
2, primes which are 1 or 7 mod 8, and squares of primes which are 3 or 5
mod 8.

These seem closely related and the numerical coincidence is compelling. Can
anyone prove (or disprove) this?

Charles Greathouse
Case Western Reserve University