A349444 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A092673 (Dirichlet inverse of A001511).

1, -1, 1, 0, 2, -1, 3, 0, 3, -2, 5, 0, 6, -3, 4, 0, 8, -3, 9, 0, 6, -5, 11, 0, 10, -6, 9, 0, 14, -4, 15, 0, 10, -8, 12, 0, 18, -9, 12, 0, 20, -6, 21, 0, 12, -11, 23, 0, 21, -10, 16, 0, 26, -9, 20, 0, 18, -14, 29, 0, 30, -15, 18, 0, 24, -10, 33, 0, 22, -12, 35, 0, 36, -18, 20, 0, 30, -12, 39, 0, 27, -20, 41, 0, 32

Offset: 1

Author

Antti Karttunen, Nov 18 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A001511, A003602, A008683, A092673, A349445 (Dirichlet inverse), A349446 (sum with it).

Cf. also A349431, A349447.

Programs

Mathematica

s[n_] := MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]]; k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, s[#]*k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

PARI

A003602(n) = (1+(n>>valuation(n,2)))/2;
A092673(n) = if(n<1, 0, moebius(n) - if( n%2, 0, moebius(n/2))); \\ From A092673
A349444(n) = sumdiv(n,d,A003602(n/d)*A092673(d));

Formula

a(n) = Sum_{d|n} A003602(n/d) * A092673(d).