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

Offset: 1

Author

Antti Karttunen, Nov 23 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A003415, A007947, A325126.

Cf. also A349394, A349612.

Programs

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 *)

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(dA007947(n/d)*A325126(d),0)); mapput(memoA325126,n,v); (v)));
A349618(n) = sumdiv(n,d,A003415(d)*A325126(n/d));

Formula

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