A308668 a(n) = Sum_{d|n} d^(n/d+n).
1, 9, 82, 1089, 15626, 287010, 5764802, 135270401, 3487315843, 100244173394, 3138428376722, 107072686593858, 3937376385699290, 155601328490478978, 6568412173896940652, 295165920677390712833, 14063084452067724991010
Offset: 1
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..385
Programs
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Mathematica
a[n_] := DivisorSum[n, #^(n/# + n) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)
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PARI
a(n) = sumdiv(n,d,d^(n/d+n));
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PARI
my(N=20, x='x+O('x^N)); Vec(x*deriv(-log(prod(k=1, N, (1-k*(k*x)^k)^(1/k))))) -
PARI
my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k+1)*x^k/(1-k^(k+1)*x^k))) \\ Seiichi Manyama, Mar 17 2021 -
Python
from sympy import divisors def A308668(n): return sum(d**(n//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*(k*x)^k)^(1/k)) = Sum_{k>=1} a(k)*x^k/k.
G.f.: Sum_{k>=1} k^(k+1) * x^k/(1 - k^(k+1) * x^k). - Seiichi Manyama, Mar 17 2021
a(n) ~ n^(n+1). - Vaclav Kotesovec, Aug 30 2025