A349619 Dirichlet convolution of A003415 with the Dirichlet inverse of A003557.

0, 1, 1, 3, 1, 3, 1, 7, 5, 5, 1, 7, 1, 7, 6, 15, 1, 9, 1, 13, 8, 11, 1, 15, 9, 13, 19, 19, 1, 14, 1, 31, 12, 17, 10, 17, 1, 19, 14, 29, 1, 20, 1, 31, 24, 23, 1, 31, 13, 25, 18, 37, 1, 27, 14, 43, 20, 29, 1, 30, 1, 31, 34, 63, 16, 32, 1, 49, 24, 34, 1, 33, 1, 37, 34, 55, 16, 38, 1, 61, 65, 41, 1, 44, 20, 43, 30, 71

Offset: 1

Author

Antti Karttunen, Nov 25 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A003415, A003557, A349340.

Cf. also A349394, A349618, A349620, A349621.

Programs

Mathematica

f1[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f1 @@@ FactorInteger[n]; f[p_, e_] := -(p - 1)^(e - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, s[#]*d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 25 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };
A349619(n) = sumdiv(n,d,A003415(n/d)*A349340(d));

Formula

a(n) = Sum_{d|n} A003415(n/d) * A349340(d).