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

1,4

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

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).

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 *)

Prog

(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));

Crossrefs

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

Cf. also A348983, A349143.

Sequence in context: A181762 A209684 A173988 * A158893 A342636 A332941

Adjacent sequences: A349120 A349121 A349122 * A349124 A349125 A349126

Keyword

nonn

Author

Antti Karttunen, Nov 08 2021

Status

approved