A368580 a(n) = Sum_{d|n and d^2 <= n} (1 + [d^2 < n]) * (2*d - 1), where [.] denote the Iverson brackets.

1, 2, 2, 5, 2, 8, 2, 8, 7, 8, 2, 18, 2, 8, 12, 15, 2, 18, 2, 22, 12, 8, 2, 32, 11, 8, 12, 22, 2, 36, 2, 22, 12, 8, 20, 43, 2, 8, 12, 40, 2, 40, 2, 22, 30, 8, 2, 54, 15, 26, 12, 22, 2, 40, 20, 48, 12, 8, 2, 72, 2, 8, 38, 37, 20, 40, 2, 22, 12, 52, 2, 84, 2, 8

Offset: 1

Author

Peter Luschny, Dec 31 2023

Keywords

• nonn

Comments

A quadruple (w, x, y, z) of nonnegative integers is a 'Bacher representation' of n if and only if n = w*x + y*z and max(w,x) < min(y,z).

A Bacher representation is 'monotone' if additionally w <= x <= y <= z.

A Bacher representation is 'degenerated' if w = 0. The weight of a Bacher representation is defined as

W(w, x, y, z) = max(1, 2*([w < x] + [y < z])).

a(n) is the sum of the weights of all degenerated monotone Bacher representations of n. The complementary sum of weights of nondegenerated monotone Bacher representations is A368581.

Examples

			Below are the monotone Bacher representations of n = 27 listed.
  W(0, 0, 1, 27) = 2;
  W(0, 0, 3,  9) = 2;
  W(0, 1, 3,  9) = 4;
  W(0, 2, 3,  9) = 4;
  W(1, 1, 2, 13) = 2;
  W(1, 2, 5,  5) = 2;
  W(1, 3, 4,  6) = 4.
Thus a(27) = 2 + 2 + 4 + 4 = 12. Adding all weights gives A368207(27) = 20.
For instance, the integers n = 6, 8, and 12 have only degenerated Bacher representation, so for these cases, a(n) = A368207(n).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• 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, A368581.

Programs

Julia

using Nemo
function A368580(n)
    sum(d * d == n ? d * 2 - 1 : d * 4 - 2
    for d in (d for d in divisors(n) if d * d <= n))
end
println([A368580(n) for n in 1:74])

Mathematica

A368580[n_]:=DivisorSum[n,(1+Boole[#^2A368580,100] (* Paolo Xausa, Jan 01 2024 *)

Formula

a(p) = 2 for all prime p.

a(n) is odd if and only if n is a square.

a(n) + A368581(n) = A368207(n).