A298825 - OEIS

%I #29 May 05 2019 11:57:11

%S 1,0,0,4,0,0,0,16,-9,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,-25,0,-54,0,0,0,0,

%T 128,0,0,0,-36,0,0,0,0,0,0,0,0,0,0,0,0,-49,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,320,0,0,0,0,0,0,0,-144,0,0,0,0,0,0,0,0,-243,0,0,0,0,0,0,0

%N Row sums of A298824.

%C Positions of nonzero entries appear to be given by A001694.

%H Antti Karttunen, <a href="/A298825/b298825.txt">Table of n, a(n) for n = 1..2500</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 From _Mats Granvik_, Mar 03 2019: (Start)

%F a(n) = n*Sum_{j=1..n} [j divides n]*A000005(n/j)*A306653(j).

%F a(n) = n*Sum_{j=1..n} [j divides n]*A000005(n/j)*Sum_{m=1..n} Sum_{k=1..n} [k divides j]*[j/k divides m]*A008683(j/k)*j/k*[k divides m + 2^p]*A008683(k)*k, p=1,2,3,4,5,...,infinity.

%F (End)

%t a[n_] := n*Sum[If[Mod[n, j] == 0,DivisorSigma[0, n/j]*1/j*Sum[Sum[If[Mod[j, k] == 0, If[Mod[m, j/k] == 0, MoebiusMu[j/k]*j/k, 0], 0]*If[Mod[m + 2, k/1] == 0, MoebiusMu[k/1]*k/1, 0], {k, 1, j}], {m, 1, j}], 0], {j, 1, n}]; a /@ Range[85] (* _Mats Granvik_, Mar 03 2019 *)

%o (PARI)

%o up_to = 256;

%o DirConv(ma,h) = { my(u = matsize(ma)[1], md = matrix(u,u)); for(n=1,u-h,for(k=1,u,md[n,k] = sumdiv(k,d,ma[n,d]*ma[n+h,k/d]))); (md); };

%o A298825list(up_to) = { my(h=2, matA = matrix(up_to+h,up_to+h,n,k,!(n%k)), matB = matrix(up_to+h,up_to+h,n,k,(!(k%n))*moebius(n)*n), matT = matA*matB, matD = DirConv(matT,2)); vector(up_to,i,sum(j=1,i,matD[j,i])); };

%o v298825 = A298825list(up_to);

%o A298825(n) = v298825[n]; \\ _Antti Karttunen_, Sep 30 2018

%Y Cf. A298674, A298824, A298826.

%K sign

%O 1,4

%A _Mats Granvik_, Jan 27 2018