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%I #40 Nov 09 2022 19:44:41
%S 1,-2,-2,2,-2,4,-2,0,0,4,-2,-4,-2,4,4,0,-2,0,-2,-4,4,4,-2,0,0,4,0,-4,
%T -2,-8,-2,0,4,4,4,0,-2,4,4,0,-2,-8,-2,-4,0,4,-2,0,0,0,4,-4,-2,0,4,0,4,
%U 4,-2,8,-2,4,0,0,4,-8,-2,-4,4,-8,-2,0,-2,4,0,-4,4,-8,-2,0,0,4,-2,8,4
%N a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n]*[n/k divides m]*A008683(n/k)*n/k*[k divides m + 2^p]*A008683(k)*k, where p can be a positive integer: 1,2,3,4,5,...
%C It appears that for p=1 and p=2, a(n) is identically the same for all n except for n equal to multiples of 16. See A306652 for comparison.
%H Antti Karttunen, <a href="/A306653/b306653.txt">Table of n, a(n) for n = 1..20000</a>
%H Mats Granvik, <a href="https://mathoverflow.net/q/308990/25104">Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture</a>
%F a(n) = 1/n * Sum_{m=1..n} Sum_{k=1..n} [k divides n]*[n/k divides m]*[k divides m + 2^p]*A008683(n/k)*n/k*A008683(k)*k, where p can be any positive integer: 1,2,3,4,5,...
%F a(n) = A000005(n)*(A008683(n) + Sum_{j=1..n} [4*A056911(j)=n]*A008836(n)*2/3) (conjectured formula verified for n=1..2500).
%F Dirichlet generating function, after Daniel Suteu in A298826 and Álvar Ibeas in A076479, appears to be: Sum_{n>=1} a(n)/n^s = (Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + Sum_{n>=2} (1/2/2^(n*(s - 1)))). - _Mats Granvik_, Apr 07 2019
%F The conjectured Dirichlet generating function simplifies to: Sum_{n>=1} a(n)/n^s = (Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + 2^(1 - s)/(2^s - 2)). - _Steven Foster Clark_, Sep 23 2022
%t (* Dirichlet Convolution. *)
%t p=1;
%t a[n_] := 1/n*Sum[Sum[If[Mod[n, k] == 0, 1, 0]*If[Mod[m, n/k] == 0, 1, 0]*If[Mod[m + 2^p, k] == 0, 1, 0]*MoebiusMu[n/k]*n/k*MoebiusMu[k]*k, {k, 1, n}], {m, 1, n}]; a /@ Range[85]
%t (* conjectured faster program *)
%t nn = 85;
%t b = 4*Select[Range[1, nn, 2], SquareFreeQ];
%t bb = Table[DivisorSigma[0, n]*(MoebiusMu[n] + Sum[If[b[[j]] == n, LiouvilleLambda[n]*2/3, 0], {j, 1, Length[b]}]), {n, 1, nn}]
%o (PARI) A306653(n) = (1/n)*sum(m=1, n, sumdiv(n, k, if( !(m%(n/k)) && !((m+(2^1))%k), n*moebius(n/k)*moebius(k),0))); \\ _Antti Karttunen_, Mar 15 2019
%Y Cf. A008683, A147848, A306652, A298825, A008683.
%K sign
%O 1,2
%A _Mats Granvik_, Mar 03 2019
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