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URL: https://oeis.org/A368580

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A368580
a(n) = Sum_{d|n and d^2 <= n} (1 + [d^2 < n]) * (2*d - 1), where [.] denote the Iverson brackets.
4
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
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
KEYWORD
nonn
AUTHOR
Peter Luschny, Dec 31 2023
STATUS
approved