A298825 Row sums of A298824.

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, 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, 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

Offset: 1

Author

Mats Granvik, Jan 27 2018

Keywords

• sign

Comments

Positions of nonzero entries appear to be given by A001694.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2500
• Mats Granvik, Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

Crossrefs

Cf. A298674, A298824, A298826.

Programs

Mathematica

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 *)

PARI

up_to = 256;
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); };
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])); };
v298825 = A298825list(up_to);
A298825(n) = v298825[n]; \\ Antti Karttunen, Sep 30 2018

Formula

From Mats Granvik, Mar 03 2019: (Start)

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

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.

(End)