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

1,7

Comments

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

Links

Table of n, a(n) for n=1..72.

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.

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.

Example

See the example in A368580.

Mathematica

t[n_]:=t[n]=Select[Divisors[n], #^2<=n&];
A368581[n_]:=Sum[If[wx<y*w, Max[1, 2(Boole[w^2<wx]+Boole[y^2<n-wx])], 0], {wx, Floor[n/2]}, {w, t[wx]}, {y, t[n-wx]}];
Array[A368581, 100] (* Paolo Xausa, Jan 02 2024 *)

Prog

(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])

Crossrefs

Cf. A368207, A368276, A368580.

Sequence in context: A053470 A308202 A329152 * A337563 A278648 A353508

Adjacent sequences: A368578 A368579 A368580 * A368582 A368583 A368584

Keyword

nonn

Author

Peter Luschny, Dec 31 2023

Status

approved