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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.

%I A368580 #22 Jan 01 2024 09:06:01
%S A368580 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,
%T A368580 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,
%U A368580 48,12,8,2,72,2,8,38,37,20,40,2,22,12,52,2,84,2,8
%N A368580 a(n) = Sum_{d|n and d^2 <= n} (1 + [d^2 < n]) * (2*d - 1), where [.] denote the Iverson brackets.
%C A368580 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).
%C A368580 A Bacher representation is 'monotone' if additionally w <= x <= y <= z.
%C A368580 A Bacher representation is 'degenerated' if w = 0. The weight of a Bacher representation is defined as
%C A368580   W(w, x, y, z) = max(1, 2*([w < x] + [y < z])).
%C A368580 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.
%H A368580 Paolo Xausa, <a href="/A368580/b368580.txt">Table of n, a(n) for n = 1..10000</a>
%H A368580 Roland Bacher, <a href="https://doi.org/10.1080/00029890.2023.2242034">A quixotic proof of Fermat's two squares theorem for prime numbers</a>, American Mathematical Monthly, Vol. 130, No. 9 (November 2023), 824-836; <a href="https://arxiv.org/abs/2210.07657">arXiv version</a>, arXiv:2210.07657 [math.NT], 2022.
%F A368580 a(p) = 2 for all prime p.
%F A368580 a(n) is odd if and only if n is a square.
%F A368580 a(n) + A368581(n) = A368207(n).
%e A368580 Below are the monotone Bacher representations of n = 27 listed.
%e A368580   W(0, 0, 1, 27) = 2;
%e A368580   W(0, 0, 3,  9) = 2;
%e A368580   W(0, 1, 3,  9) = 4;
%e A368580   W(0, 2, 3,  9) = 4;
%e A368580   W(1, 1, 2, 13) = 2;
%e A368580   W(1, 2, 5,  5) = 2;
%e A368580   W(1, 3, 4,  6) = 4.
%e A368580 Thus a(27) = 2 + 2 + 4 + 4 = 12. Adding all weights gives A368207(27) = 20.
%e A368580 For instance, the integers n = 6, 8, and 12 have only degenerated Bacher representation, so for these cases, a(n) = A368207(n).
%t A368580 A368580[n_]:=DivisorSum[n,(1+Boole[#^2<n])(2#-1)&,#^2<=n&];
%t A368580 Array[A368580,100] (* _Paolo Xausa_, Jan 01 2024 *)
%o A368580 (Julia)
%o A368580 using Nemo
%o A368580 function A368580(n)
%o A368580     sum(d * d == n ? d * 2 - 1 : d * 4 - 2
%o A368580     for d in (d for d in divisors(n) if d * d <= n))
%o A368580 end
%o A368580 println([A368580(n) for n in 1:74])
%Y A368580 Cf. A368207, A368276, A368581.
%K A368580 nonn
%O A368580 1,2
%A A368580 _Peter Luschny_, Dec 31 2023