A349173 Dirichlet convolution of A003415 with A003959, where A003415 is the arithmetic derivative and A003959 is fully multiplicative with a(p) = (p+1).

0, 1, 1, 7, 1, 12, 1, 33, 10, 16, 1, 68, 1, 20, 18, 131, 1, 87, 1, 96, 22, 28, 1, 296, 16, 32, 67, 124, 1, 167, 1, 473, 30, 40, 26, 449, 1, 44, 34, 428, 1, 215, 1, 180, 147, 52, 1, 1128, 22, 171, 42, 208, 1, 510, 34, 560, 46, 64, 1, 881, 1, 68, 187, 1611, 38, 311, 1, 264, 54, 295, 1, 1871, 1, 80, 203, 292, 38, 359

Offset: 1

Author

Antti Karttunen, Nov 09 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A003415, A003959, A349129, A349133, A349170, A349171, A349172, A349356.

Programs

Mathematica

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a1[1] = 0; a1[n_] := n*Plus @@ (f1 @@@ FactorInteger[n]); a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, a1[#] * a2[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349173(n) = sumdiv(n,d,A003415(d)*A003959(n/d));

Formula

a(n) = Sum_{d|n} A003415(d) * A003959(n/d).

a(n) = Sum_{d|n} A349133(d) * A349356(n/d). - Antti Karttunen, Nov 16 2021

For all n >= 1, a(n) >= A349133(n).