A349356 - OEIS

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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

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OFFSET

1,2

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

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

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)

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.