A349356 Dirichlet convolution of A003959 with A097945 (Dirichlet inverse of A003958), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

1, 2, 2, 6, 2, 4, 2, 18, 8, 4, 2, 12, 2, 4, 4, 54, 2, 16, 2, 12, 4, 4, 2, 36, 12, 4, 32, 12, 2, 8, 2, 162, 4, 4, 4, 48, 2, 4, 4, 36, 2, 8, 2, 12, 16, 4, 2, 108, 16, 24, 4, 12, 2, 64, 4, 36, 4, 4, 2, 24, 2, 4, 16, 486, 4, 8, 2, 12, 4, 8, 2, 144, 2, 4, 24, 12, 4, 8, 2, 108, 128, 4, 2, 24, 4, 4, 4, 36, 2, 32, 4, 12, 4

Offset: 1

Author

Antti Karttunen, Nov 16 2021

Keywords

• nonn
• mult

Comments

In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003958 with factor A003959. For example, convolving this with A349133 produces A349173.

Links

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

Crossrefs

Cf. A003958, A003959, A097945, A349355 (Dirichlet inverse), A349357 (sum with it).

Cf. also A349133, A349173, A349381.

Programs

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 *)

PARI

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A097945(n) = (moebius(n)*eulerphi(n)); \\ Also Dirichlet inverse of A003958.
A349356(n) = sumdiv(n,d,A003959(n/d)*A097945(d));

Formula

a(n) = Sum_{d|n} A003959(n/d) * A097945(d).

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