A349447 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A326937 (Dirichlet inverse of A000265).

1, 0, -1, 0, -2, 0, -3, 0, -1, 0, -5, 0, -6, 0, 4, 0, -8, 0, -9, 0, 6, 0, -11, 0, -2, 0, -1, 0, -14, 0, -15, 0, 10, 0, 12, 0, -18, 0, 12, 0, -20, 0, -21, 0, 4, 0, -23, 0, -3, 0, 16, 0, -26, 0, 20, 0, 18, 0, -29, 0, -30, 0, 6, 0, 24, 0, -33, 0, 22, 0, -35, 0, -36, 0, 4, 0, 30, 0, -39, 0, -1, 0, -41, 0, 32, 0, 28

Offset: 1

Author

Antti Karttunen, Nov 19 2021

Keywords

• sign

Comments

Dirichlet convolution of this sequence with A264740 is A349371.

Links

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

Crossrefs

Cf. A000265, A003602, A326937, A349448 (Dirichlet inverse).

Cf. also A264740, A349371, A349431, A349444.

Programs

Mathematica

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

PARI

A003602(n) = (1+(n>>valuation(n,2)))/2;
A006519(n) = (1<A055615(n) = (n*moebius(n));
A326937(n) = (A055615(n)/A006519(n));
A349447(n) = sumdiv(n,d,A003602(d)*A326937(n/d));

Formula

a(n) = Sum_{d|n} A003602(d) * A326937(n/d).