A066108 Sum n^d over all divisors of n.

1, 6, 30, 276, 3130, 46914, 823550, 16781384, 387421227, 10000100110, 285311670622, 8916103456860, 302875106592266, 11112006930971730, 437893890381622140, 18446744078004584720, 827240261886336764194, 39346408075494930884190, 1978419655660313589123998

Offset

1,2

Comments

This is neither the Moebius transform nor the inverse Moebius transform of n^n, although it is close to them.

Links

Harry J. Smith, Table of n, a(n) for n = 1..100

Formula

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

a(n) ~ n^n. - Vaclav Kotesovec, Jun 05 2021

Conjectured g.f.: Sum_{m>=1} (m^m*Sum_{k=1..m}(x^(k*m)*Sum_{j=0..k}((-1)^j*(k - j)^m*binomial(m + 1, j)))/(1 - x^m)^(m + 1)), where the inner summation is the Triangle of Eulerian numbers A008292. - Miles Wilson, Jan 12 2025

Example

n = 12: a(12) = A066106(12) = 8916103456860 = 8916100448256+2985984+20736+1728+144+12.
For comparison: M-transform of n^n at 12 = 8916100401348 = 8916100448256-46656-256+0+4+0 = A062793(12);
Inverse M-transform of n^n at 12 = 8916100495200 = 8916100448256+46656+256+27+4+1 = A062796(12).

Maple

A066108 := n -> add(n^d, d = NumberTheory[Divisors](n)); seq(A066108(n), n = 1 .. 19) - Miles Wilson, Jan 12 2025

Mathematica

Table[Sum[n^d, {d, Divisors@ n}], {n, 19}] (* Michael De Vlieger, Dec 20 2015 *)

Prog

(PARI) a(n)=sumdiv(n, d, n^d );  /* Joerg Arndt, Oct 07 2012 */

Crossrefs

Cf. A062793, A062796.

Sequence in context: A343574 A051821 A099031 * A205339 A215288 A295611

Adjacent sequences: A066105 A066106 A066107 * A066109 A066110 A066111

Keyword

nonn

Author

Labos Elemer, Dec 05 2001

Status

approved