A308756 -id:A308756 - cp's OEIS frontend

A308696 a(n) = Sum_{d|n} d^(2*d).

1, 17, 730, 65553, 9765626, 2176783082, 678223072850, 281474976776209, 150094635296999851, 100000000000009765642, 81402749386839761113322, 79496847203390846310290154, 91733330193268616658399616010, 123476695691247935826908004929122

Offset: 1

Seiichi Manyama, Jun 17 2019

nonn

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

Column k=2 of A308698.

Cf. A073705, A308753, A308756.

a[n_] := DivisorSum[n, #^(2*#) &]; Array[a, 14] (* Amiram Eldar, May 09 2021 *)

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

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

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

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

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

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A308696 a(n) = Sum_{d|n} d^(2*d).

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

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Author

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Mathematica

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