A349355 Dirichlet convolution of A003958 with A063441 (Dirichlet inverse of A003959), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

1, -2, -2, -2, -2, 4, -2, -2, -4, 4, -2, 4, -2, 4, 4, -2, -2, 8, -2, 4, 4, 4, -2, 4, -8, 4, -8, 4, -2, -8, -2, -2, 4, 4, 4, 8, -2, 4, 4, 4, -2, -8, -2, 4, 8, 4, -2, 4, -12, 16, 4, 4, -2, 16, 4, 4, 4, 4, -2, -8, -2, 4, 8, -2, 4, -8, -2, 4, 4, -8, -2, 8, -2, 4, 16, 4, 4, -8, -2, 4, -16, 4, -2, -8, 4, 4, 4, 4, -2, -16

Offset

1,2

Comments

Multiplicative because both A003958 and A063441 are.

In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003959 with factor A003958. For example, convolving this with A003968 (the Möbius transform of A003959) produces A003966, the Möbius transform of A003958.

Links

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

Wikipedia, Dirichlet convolution

Formula

a(n) = Sum_{d|n} A003958(n/d) * A063441(d).

Multiplicative with a(p^e) = -2*(p-1)^(e-1). - Amiram Eldar, Nov 16 2021

Mathematica

f[p_, e_] := -2*(p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

Prog

(PARI)
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A063441(n) = (moebius(n)*sigma(n)); \\ Also Dirichlet inverse of A003959.
A349355(n) = sumdiv(n, d, A003958(n/d)*A063441(d));

Crossrefs

Cf. A003958, A003959, A003966, A003968, A063441, A349356 (Dirichlet inverse), A349357 (sum with it).

Cf. also A349382.

Sequence in context: A081755 A366538 A366536 * A353589 A237709 A250200

Adjacent sequences: A349352 A349353 A349354 * A349356 A349357 A349358

Keyword

sign,mult

Author

Antti Karttunen, Nov 16 2021

Status

approved