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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,... 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

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

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

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}]

PROG

(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

CROSSREFS

Cf. A008683, A147848, A306652, A298825, A008683.

Sequence in context: A092904 A231883 A062816 * A132003 A122857 A109810

Adjacent sequences:  A306650 A306651 A306652 * A306654 A306655 A306656

KEYWORD

sign

AUTHOR

Mats Granvik, Mar 03 2019

STATUS

approved

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Last modified July 31 22:33 EDT 2021. Contains 346377 sequences. (Running on oeis4.)