A066108 - OEIS

%I #36 Jan 23 2025 15:37:19

%S 1,6,30,276,3130,46914,823550,16781384,387421227,10000100110,

%T 285311670622,8916103456860,302875106592266,11112006930971730,

%U 437893890381622140,18446744078004584720,827240261886336764194,39346408075494930884190,1978419655660313589123998

%N Sum n^d over all divisors of n.

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

%H Harry J. Smith, <a href="/A066108/b066108.txt">Table of n, a(n) for n = 1..100</a>

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

%F a(n) ~ n^n. - _Vaclav Kotesovec_, Jun 05 2021

%F 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

%e n = 12: a(12) = A066106(12) = 8916103456860 = 8916100448256+2985984+20736+1728+144+12.

%e For comparison: M-transform of n^n at 12 = 8916100401348 = 8916100448256-46656-256+0+4+0 = A062793(12);

%e Inverse M-transform of n^n at 12 = 8916100495200 = 8916100448256+46656+256+27+4+1 = A062796(12).

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

%t Table[Sum[n^d, {d, Divisors@ n}], {n, 19}] (* _Michael De Vlieger_, Dec 20 2015 *)

%o (PARI) a(n)=sumdiv(n,d, n^d );  /* _Joerg Arndt_, Oct 07 2012 */

%Y Cf. A062793, A062796.

%K nonn,changed

%O 1,2

%A _Labos Elemer_, Dec 05 2001