A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

0, 1, 1, 0, 1, 0, 1, -1, -1, 0, 1, -2, 1, 0, 0, -2, 1, -6, 1, -2, 0, 0, 1, -2, -3, 0, -3, -2, 1, 0, 1, -3, 0, 0, 0, 2, 1, 0, 0, -2, 1, 0, 1, -2, -6, 0, 1, -2, -5, -20, 0, -2, 1, -6, 0, -2, 0, 0, 1, 0, 1, 0, -6, -4, 0, 0, 1, -2, 0, 0, 1, 8, 1, 0, -20, -2, 0, 0, 1, -2, -5, 0, 1, 0, 0, 0, 0, -2, 1, 0, 0, -2, 0, 0, 0

Offset: 1

Author

Antti Karttunen, Nov 18 2021

Keywords

• sign

Comments

Dirichlet convolution of this sequence with A000010 (Euler phi) is A346485.

Links

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

Crossrefs

Cf. A003415, A003557, A055615, A342001.

Cf. A346485, A347234, A347235, A347395, A347954, A347959, A347961, A347963 for Dirichlet convolutions of A342001 with other sequences.

Cf. also A349394.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A055615(n) = (n*moebius(n));
A349396(n) = sumdiv(n,d,A342001(n/d)*A055615(d));

Formula

a(n) = Sum_{d|n} A055615(d) * A342001(n/d).