A347130 a(n) = Sum_{d|n} d * A003415(n/d), where A003415 is the arithmetic derivative.

0, 1, 1, 6, 1, 10, 1, 24, 9, 14, 1, 48, 1, 18, 16, 80, 1, 63, 1, 72, 20, 26, 1, 176, 15, 30, 54, 96, 1, 124, 1, 240, 28, 38, 24, 270, 1, 42, 32, 272, 1, 164, 1, 144, 117, 50, 1, 560, 21, 135, 40, 168, 1, 324, 32, 368, 44, 62, 1, 552, 1, 66, 153, 672, 36, 244, 1, 216, 52, 236, 1, 936, 1, 78, 165, 240, 36, 284, 1, 880

Offset: 1

Author

Antti Karttunen, Aug 23 2021

Keywords

• nonn

Comments

Dirichlet convolution of the identity function (A000027) with the arithmetic derivative of n (A003415).

Dirichlet convolution of Euler phi (A000010) with A319684.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10080
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Inverse Möbius transform of A347131.

Cf. A000010, A000027, A003415, A003557, A319684, A347129.

Cf. also A300251, A305809, A319684, A347132, A347133.

Programs

Mathematica

Table[DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &], {n, 80}] (* Michael De Vlieger, Oct 21 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));

Formula

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

a(n) = Sum_{d|n} A000010(n/d) * A319684(d).

a(n) = Sum_{d|n} A347131(d).

a(n) = A003557(n) * A347129(n).