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

1,2

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.

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.

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).

Example

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).

Mathematica

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

Prog

(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])

Crossrefs

Cf. A368207, A368276, A368581.

Sequence in context: A336924 A025124 A030996 * A309208 A293225 A353851

Adjacent sequences: A368577 A368578 A368579 * A368581 A368582 A368583

Keyword

nonn

Author

Peter Luschny, Dec 31 2023

Status

approved