A308668 - OEIS

%I #29 Jun 19 2022 15:23:00

%S 1,9,82,1089,15626,287010,5764802,135270401,3487315843,100244173394,

%T 3138428376722,107072686593858,3937376385699290,155601328490478978,

%U 6568412173896940652,295165920677390712833,14063084452067724991010

%N a(n) = Sum_{d|n} d^(n/d+n).

%H Seiichi Manyama, <a href="/A308668/b308668.txt">Table of n, a(n) for n = 1..385</a>

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

%F G.f.: Sum_{k>=1} k^(k+1) * x^k/(1 - k^(k+1) * x^k). - _Seiichi Manyama_, Mar 17 2021

%t a[n_] := DivisorSum[n, #^(n/# + n) &]; Array[a, 20] (* _Amiram Eldar_, Mar 17 2021 *)

%o (PARI) a(n) = sumdiv(n,d,d^(n/d+n));

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

%o (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

%o (Python)

%o from sympy import divisors

%o def A308668(n): return sum(d**(n//d+n) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jun 19 2022

%Y Diagonal of A308502.

%Y Cf. A152211, A294956, A308594.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jun 16 2019