A349123 a(n) = Sum_{d|n} A038040(n/d) * A003415(d), where A038040(n) = n*tau(n), and A003415 is the arithmetic derivative of n.

0, 1, 1, 8, 1, 15, 1, 40, 12, 21, 1, 96, 1, 27, 24, 160, 1, 126, 1, 144, 30, 39, 1, 440, 20, 45, 90, 192, 1, 279, 1, 560, 42, 57, 36, 720, 1, 63, 48, 680, 1, 369, 1, 288, 234, 75, 1, 1680, 28, 270, 60, 336, 1, 810, 48, 920, 66, 93, 1, 1656, 1, 99, 306, 1792, 54, 549, 1, 432, 78, 531, 1, 3120, 1, 117, 330, 480, 54, 639

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Comments

This sequence is the Dirichlet convolution of at least the following pairs of sequences:

- A003415 (the arithmetic derivative) with A038040,

- A000027 (the identity function) with A347130,

- A000203 (sigma) with A347131,

- A018804 with A319684,

- A060640 with A300251.

Links

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

Crossrefs

Cf. A000027, A000203, A003415, A003557, A018804, A060640, A300251, A319684, A347130, A349124.

Cf. also A348983, A349143.

Programs

Mathematica

d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#]*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A038040(n) = (n*numdiv(n));
A349123(n) = sumdiv(n,d,A038040(d)*A003415(n/d));

Formula

a(n) = Sum_{d|n} A038040(d) * A003415(n/d).

a(n) = Sum_{d|n} d * A347130(n/d).

a(n) = Sum_{d|n} A000203(d) * A347131(n/d).

a(n) = Sum_{d|n} A018804(d) * A319684(n/d).

a(n) = Sum_{d|n} A060640(d) * A300251(n/d).

For all n >= 1, A348983(n) <= a(n) <= A349143(n).

a(n) = A003557(n) * A349124(n).