A308755 -id:A308755 - cp's OEIS frontend

A308753 a(n) = Sum_{d|n} d^(2*(d-1)).

1, 5, 82, 4101, 390626, 60466262, 13841287202, 4398046515205, 1853020188851923, 1000000000000390630, 672749994932560009202, 552061438912436478063702, 542800770374370512771595362, 629983141281877223617054459942

Offset: 1

Seiichi Manyama, Jun 22 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..215

Column k=2 of A308701.

Cf. A283533, A308692, A308696, A308755, A308756.

a[n_] := DivisorSum[n, #^(2*(# - 1)) &]; Array[a, 14] (* Amiram Eldar, May 08 2021 *)

PARI
```
{a(n) = sumdiv(n, d, d^(2*(d-1)))}
```

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(2*k-3)))))

N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(2*(k-1))*x^k/(1-x^k)))

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(2*k-3))) = Sum_{k>=1} a(k)*x^k/k.

G.f.: Sum_{k>=1} k^(2*(k-1)) * x^k/(1 - x^k).

A308763 a(n) = Sum_{d|n} d^(n-2).

Original entry on oeis.org

1, 2, 4, 21, 126, 1394, 16808, 266305, 4785157, 100390882, 2357947692, 61978939050, 1792160394038, 56707753666594, 1946196290656824, 72061992352890881, 2862423051509815794, 121441386937936123331, 5480386857784802185940, 262145000003883417004506

Offset: 1

Seiichi Manyama, Jun 23 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A023887, A082245, A294645, A294810, A308755.

a[n_] := DivisorSum[n, #^(n - 2) &]; Array[a, 20] (* Amiram Eldar, May 08 2021 *)

PARI
```
{a(n) = sigma(n, n-2)}
```

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^(1/k^3)))))

N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(k-2)*x^k/(1-(k*x)^k)))

L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^(1/k^3)) = Sum_{k>=1} a(k)*x^k/k.

G.f.: Sum_{k>=1} k^(k-2) * x^k/(1 - (k*x)^k).

cp's OEIS Frontend

A308753 a(n) = Sum_{d|n} d^(2*(d-1)).

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A308763 a(n) = Sum_{d|n} d^(n-2).

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Author

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Mathematica

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