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A308668 a(n) = Sum_{d|n} d^(n/d+n).

%I A308668 #31 Aug 30 2025 10:39:07
%S A308668 1,9,82,1089,15626,287010,5764802,135270401,3487315843,100244173394,
%T A308668 3138428376722,107072686593858,3937376385699290,155601328490478978,
%U A308668 6568412173896940652,295165920677390712833,14063084452067724991010
%N A308668 a(n) = Sum_{d|n} d^(n/d+n).
%H A308668 Seiichi Manyama, <a href="/A308668/b308668.txt">Table of n, a(n) for n = 1..385</a>
%F A308668 L.g.f.: -log(Product_{k>=1} (1 - k*(k*x)^k)^(1/k)) = Sum_{k>=1} a(k)*x^k/k.
%F A308668 G.f.: Sum_{k>=1} k^(k+1) * x^k/(1 - k^(k+1) * x^k). - _Seiichi Manyama_, Mar 17 2021
%F A308668 a(n) ~ n^(n+1). - _Vaclav Kotesovec_, Aug 30 2025
%t A308668 a[n_] := DivisorSum[n, #^(n/# + n) &]; Array[a, 20] (* _Amiram Eldar_, Mar 17 2021 *)
%o A308668 (PARI) a(n) = sumdiv(n,d,d^(n/d+n));
%o A308668 (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 A308668 (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 A308668 (Python)
%o A308668 from sympy import divisors
%o A308668 def A308668(n): return sum(d**(n//d+n) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jun 19 2022
%Y A308668 Diagonal of A308502.
%Y A308668 Cf. A152211, A294956, A308594.
%K A308668 nonn,changed
%O A308668 1,2
%A A308668 _Seiichi Manyama_, Jun 16 2019