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

1,4

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

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).

Mathematica

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

Prog

(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));

Crossrefs

Inverse Möbius transform of A347131.

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

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

Sequence in context: A097186 A329047 A363316 * A070533 A336475 A082744

Adjacent sequences: A347127 A347128 A347129 * A347131 A347132 A347133

Keyword

nonn

Author

Antti Karttunen, Aug 23 2021

Status

approved