A349618 - OEIS

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A349618

Dirichlet convolution of arithmetic derivative with A325126 [Dirichlet inverse of rad(n)].

0, 1, 1, 2, 1, 0, 1, 6, 3, 0, 1, 2, 1, 0, 0, 14, 1, 6, 1, 2, 0, 0, 1, 2, 5, 0, 15, 2, 1, 0, 1, 34, 0, 0, 0, 18, 1, 0, 0, 2, 1, 0, 1, 2, 6, 0, 1, 6, 7, 20, 0, 2, 1, 6, 0, 2, 0, 0, 1, 0, 1, 0, 6, 78, 0, 0, 1, 2, 0, 0, 1, 42, 1, 0, 20, 2, 0, 0, 1, 6, 51, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 10, 1, 42, 6, 50, 1, 0, 1, 2, 0

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OFFSET

1,4

LINKS

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

FORMULA

a(n) = Sum_{d|n} A003415(d) * A325126(n/d).

MATHEMATICA

f[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f @@@ FactorInteger[n]; f2[p_, e_] := -p*(1 - p)^(e - 1); s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#]*s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947

memoA325126 = Map();

A325126(n) = if(1==n, 1, my(v); if(mapisdefined(memoA325126, n, &v), v, v = -sumdiv(n, d, if(d<n, A007947(n/d)*A325126(d), 0)); mapput(memoA325126, n, v); (v)));

A349618(n) = sumdiv(n, d, A003415(d)*A325126(n/d));

CROSSREFS

Cf. A003415, A007947, A325126.

Cf. also A349394, A349612.

Sequence in context: A108723 A291584 A352451 * A321615 A011126 A266854

Adjacent sequences: A349615 A349616 A349617 * A349619 A349620 A349621

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 23 2021

STATUS

approved