A306653 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,...
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, -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, 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
Offset: 1
Keywords
Links
- Antti Karttunen, Table of n, a(n) for n = 1..20000
- Mats Granvik, Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture
Programs
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Mathematica
(* Dirichlet Convolution. *) p=1; 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] (* conjectured faster program *) nn = 85; b = 4*Select[Range[1, nn, 2], SquareFreeQ]; bb = Table[DivisorSigma[0, n]*(MoebiusMu[n] + Sum[If[b[[j]] == n, LiouvilleLambda[n]*2/3, 0], {j, 1, Length[b]}]), {n, 1, nn}] -
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
Formula
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,...
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).
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
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
Comments