A349441 Dirichlet convolution of A057521 (powerful part of n) with A055615 (Dirichlet inverse of n).

1, -1, -2, 2, -4, 2, -6, 0, 6, 4, -10, -4, -12, 6, 8, 0, -16, -6, -18, -8, 12, 10, -22, 0, 20, 12, 0, -12, -28, -8, -30, 0, 20, 16, 24, 12, -36, 18, 24, 0, -40, -12, -42, -20, -24, 22, -46, 0, 42, -20, 32, -24, -52, 0, 40, 0, 36, 28, -58, 16, -60, 30, -36, 0, 48, -20, -66, -32, 44, -24, -70, 0, -72, 36, -40, -36

Offset: 1

Author

Antti Karttunen, Nov 18 2021

Keywords

• sign
• mult
• look

Comments

Multiplicative because A055615 and A057521 are.

Convolving this with Euler phi (A000010) produces A349379.

Links

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

Crossrefs

Cf. A055615, A057521, A349442 (Dirichlet inverse), A349443 (sum with it).

Cf. also A097945, A349379.

Programs

Mathematica

f[p_, e_] := Which[e > 2, 0, e == 2, p^2 - p, e == 1, 1 - p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

PARI

A055615(n) = (n*moebius(n));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349441(n) = sumdiv(n,d,A057521(n/d)*A055615(d));

Formula

a(n) = Sum_{d|n} A057521(n/d) * A055615(d).

Multiplicative with a(p^e) = 1 - p is e = 1, p^2 - p if e = 2, and 0 otherwise. - Amiram Eldar, Nov 19 2021