A349133 Dirichlet convolution of A003415 with A003958, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

0, 1, 1, 5, 1, 8, 1, 17, 8, 12, 1, 32, 1, 16, 14, 49, 1, 43, 1, 52, 18, 24, 1, 100, 14, 28, 43, 72, 1, 87, 1, 129, 26, 36, 22, 151, 1, 40, 30, 168, 1, 119, 1, 112, 91, 48, 1, 276, 20, 103, 38, 132, 1, 194, 30, 236, 42, 60, 1, 323, 1, 64, 123, 321, 34, 183, 1, 172, 50, 183, 1, 443, 1, 76, 131, 192, 34, 215, 1, 472

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, A003958, A349129, A349130, A349131, A349132, A349173.

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]));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349133(n) = sumdiv(n,d,A003415(d)*A003958(n/d));

Formula

a(n) = Sum_{d|n} A003415(d) * A003958(n/d).

For all n >= 1, a(n) <= A349173(n).