A119758 Numerator of Sum_{k=1..n} k^n/n^k.

1, 3, 20, 225, 3789, 89341, 2821552, 115377921, 5939637425, 375840753541, 28641787322796, 2583828842108449, 271949027324094925, 32986652806128680205, 4563200871898056653504, 713455071424061222336513

Offset: 1

Author

Alexander Adamchuk, Jun 18 2006, Jun 25 2006

Keywords

• frac
• nonn

Comments

a(p-1) is divisible by prime p>2. a(p) is divisible by ((p+1)/2)^2 for prime p>2.

Denominator of Sum[k^n/n^k,{k,1,n}] is equal to n^(n-1) = A000169(n). - Alexander Adamchuk, Jun 27 2006

Crossrefs

Cf. A023037, A031971.

Cf. A000169.

Programs

Mathematica

Table[Numerator[Sum[k^n/n^k,{k,1,n}]],{n,1,20}]
Table[Sum[k^n/n^k,{k,1,n}]*n^(n-1),{n,1,50}] (* Alexander Adamchuk, Jun 27 2006 *)

PARI

a(n) = numerator(prod(k=2, n, 1-1/(prime(k)-1)^2)); \\ Michel Marcus, May 31 2022

Formula

a(n) = numerator(Sum_{k=1..n} k^n/n^k).

a(n) = n^(n-1)*(Sum_{k=1..n} k^n/n^k). - Alexander Adamchuk, Jun 27 2006

a(2m) is divisible by 2m+1 for integer m>0. a(2m-1) is divisible by m^2 for integer m>0. - Alexander Adamchuk, Jun 27 2006