A349124 a(n) = A349123(n) / A003557(n), where A349123 is the Dirichlet convolution of the arithmetic derivative with n*tau(n).

0, 1, 1, 4, 1, 15, 1, 10, 4, 21, 1, 48, 1, 27, 24, 20, 1, 42, 1, 72, 30, 39, 1, 110, 4, 45, 10, 96, 1, 279, 1, 35, 42, 57, 36, 120, 1, 63, 48, 170, 1, 369, 1, 144, 78, 75, 1, 210, 4, 54, 60, 168, 1, 90, 48, 230, 66, 93, 1, 828, 1, 99, 102, 56, 54, 549, 1, 216, 78, 531, 1, 260, 1, 117, 66, 240, 54, 639, 1, 330, 20, 129

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A003415, A003557, A038040, A349123.

Cf. also A347129.

Programs

Mathematica

f[p_, e_] := p^(e - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#]*(n/#)*DivisorSigma[0, n/#] &] / s[n]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

PARI

A003557(n) = (n/factorback(factorint(n)[, 1]));
A349124(n) = (A349123(n) / A003557(n)); \\ Needs also code from A349123.

Formula

a(n) = A349123(n) / A003557(n).