A308594 a(n) = Sum_{d|n} d^(d+n).

1, 17, 730, 65601, 9765626, 2176802276, 678223072850, 281474993488897, 150094635297530563, 100000000030517582222, 81402749386839761113322, 79496847203492408399442540, 91733330193268616658399616010, 123476695691248494372093865205800

Offset: 1

Author

Seiichi Manyama, Jun 09 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..214

Crossrefs

Cf. A062796, A294956, A308668.

Programs

Mathematica

sp[n_]:=Module[{d=Divisors[n]},Table[d[[k]]^(d[[k]]+n),{k,Length[ d]}]] // Total; Array[sp,15] (* Harvey P. Dale, Jan 02 2020 *)
a[n_] := DivisorSum[n, #^(# + n) &]; Array[a, 14] (* Amiram Eldar, May 11 2021 *)

PARI

a(n) = sumdiv(n, d, d^(d+n));

PARI

my(N=20, x='x+O('x^N)); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^(k^(k-1))))))

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k^2*x)^k/(1-(k*x)^k))) \\ Seiichi Manyama, Mar 16 2021

Python

from sympy import divisors
def A308594(n): return sum(d**(d+n) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

Formula

L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^(k^(k-1))) = Sum_{k>=1} a(k)*x^k/k.

G.f.: Sum_{k>=1} (k^2 * x)^k/(1 - (k * x)^k). - Seiichi Manyama, Mar 16 2021