[math-fun] Re: even Q(n)

[Christian G.Bower]

So Dean Hickerson's post settles the issue for A000009(n) == 2 (mod 4)

I submitted the c(n) sequence as A115323 with a generous supply of
comments lifted from Dean's post.

I wonder if a similar result can be derived for A001935(n) == 2 (mod 4)

I created a function similar to Dean's c(n), I'll call it e(n):

Let e(n) be the number of solutions of

    8*n+1 = x^2 + 8*y^2,  x>0.

Like c(n), e(n) appears to have the property that

e(n) == 2 (mod 4) iff A001935(n) == 2 (mod 4)

If 8*n+1 is not a square or if sqrt(8*n+1) == 1 or 7 (mod 8), then

    A001935(n) == e(n) (mod 4).

If sqrt(8*n+1) == 3 or 5 (mod 8)  then

    A001935(n) == e(n) + 2 (mod 4).

but I don't know enough about quadratic equations to prove this or to
show that all n such that 8*n+1 is prime have a value of 2 or to
show how this function can be calculated from the prime factorization.

Christian