cp's OEIS Frontend

A368581 The sum of weights of nondegenerated monotone Bacher representations of n.

%I A368581 #14 Jan 02 2024 06:07:17
%S A368581 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,
%T A368581 10,12,19,10,8,17,22,16,9,19,18,20,20,11,28,22,16,20,21,24,27,25,26,
%U A368581 16,24,28,37,28,16,29,40,24,34,22,34,32,41,36,28,34,28
%N A368581 The sum of weights of nondegenerated monotone Bacher representations of n.
%C A368581 For the definition of 'Bacher representation' and related notions, see the comments in A368580.
%H A368581 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 A368581 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.)
%F A368581 a(n) + A368580(n) = A368207(n).
%F A368581 a(p) = (p + 1) / 2 - 2 for all odd prime p.
%e A368581 See the example in A368580.
%t A368581 t[n_]:=t[n]=Select[Divisors[n],#^2<=n&];
%t A368581 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]}];
%t A368581 Array[A368581,100] (* _Paolo Xausa_, Jan 02 2024 *)
%o A368581 (Julia)
%o A368581 using Nemo
%o A368581 function A368581(n::Int)
%o A368581     t(n) = (d for d in divisors(n) if d * d <= n)
%o A368581     c(y, w, wx) = max(1, 2 * (Int(w * w < wx) + Int(y * y < n - wx)))
%o A368581     sum(sum(sum(c(y, w, wx) for y in t(n - wx) if wx < y * w; init=0)
%o A368581     for w in t(wx)) for wx in 1:div(n, 2); init=0)
%o A368581 end
%o A368581 println([A368581(n) for n in 1:72])
%Y A368581 Cf. A368207, A368276, A368580.
%K A368581 nonn
%O A368581 1,7
%A A368581 _Peter Luschny_, Dec 31 2023