A368581 The sum of weights of nondegenerated monotone Bacher representations of n.

0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 5, 4, 2, 4, 7, 2, 8, 5, 4, 10, 10, 2, 9, 13, 8, 8, 13, 8, 14, 10, 12, 19, 10, 8, 17, 22, 16, 9, 19, 18, 20, 20, 11, 28, 22, 16, 20, 21, 24, 27, 25, 26, 16, 24, 28, 37, 28, 16, 29, 40, 24, 34, 22, 34, 32, 41, 36, 28, 34, 28

Offset: 1

Author

Peter Luschny, Dec 31 2023

Keywords

• nonn

Comments

For the definition of 'Bacher representation' and related notions, see the comments in A368580.

Examples

			See the example in A368580.
		

Links

• Roland Bacher, A quixotic proof of Fermat's two squares theorem for prime numbers, American Mathematical Monthly, Vol. 130, No. 9 (November 2023), 824-836; arXiv version, arXiv:2210.07657 [math.NT], 2022.

Crossrefs

Cf. A368207, A368276, A368580.

Programs

Julia

using Nemo
function A368581(n::Int)
    t(n) = (d for d in divisors(n) if d * d <= n)
    c(y, w, wx) = max(1, 2 * (Int(w * w < wx) + Int(y * y < n - wx)))
    sum(sum(sum(c(y, w, wx) for y in t(n - wx) if wx < y * w; init=0)
    for w in t(wx)) for wx in 1:div(n, 2); init=0)
end
println([A368581(n) for n in 1:72])

Mathematica

t[n_]:=t[n]=Select[Divisors[n],#^2<=n&];
A368581[n_]:=Sum[If[wxA368581,100] (* Paolo Xausa, Jan 02 2024 *)

Formula

a(n) = Sum_{k in K} Sum_{w in W} Sum_{y in Y} max(1, 2*([w^2 < k] + [y^2 < n - k])), where the square brackets denote Iverson brackets and k in K <=> 1 <= k <= floor(n/2), w in W <=> w|k and w^2 <= k, and y in Y <=> y|n-k and y^2 <= n-k and k < y*w. (See the Julia implementation.)

a(n) + A368580(n) = A368207(n).

a(p) = (p + 1) / 2 - 2 for all odd prime p.