[seqfan] Sums of Two Terms

[Frank Adams-Watters]

I've been interested for some time in the question of finding sequences that grow on the order of n^2, such that every positive integer is the sum of two members of the sequence.

A126684 has this property, but not in a uniform way. That is, there are c1 and c2 s.t. in the limit c1*n^2 <= A126684(n) <= c2*n^2; but c1 and c2 are not the same: we don't have A126684(n) = c*n^2 + o(n^2).

I recently added the sequence A256435: first differences of numbers that are the sum of two squares. It is  not hard to show that this sequence is unbounded. This means that no sequence with the requisite property can be a(n) = c*n^2 + O(1), because then the sum of two terms would be c*(n1^2+n2^2) + O(1), and eventually the gaps between numbers of this form will be large enough to skip some integers.

It seems to me that if we can get some results about how fast A256435 grows, we should be able to sharpen this result.

So this is my question: how fast does A256435 grow? In particular, I want to show that A256435(n) = Omega(f(n)) for some suitable function f. (I.e., infinitely often, A256435(n) >= c*f(n) for some value of c (which can then be folded in the definition of f)).

Franklin T. Adams-Watters