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

Author

Labos Elemer, Dec 05 2001

Keywords

• nonn

Comments

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

Examples

			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).
		

Links

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

Crossrefs

Cf. A062793, A062796.

Programs

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 *)

PARI

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

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