A308675 -id:A308675 - cp's OEIS frontend

A308670 a(n) = Sum_{d|n} d^(d*n).

1, 17, 19684, 4294967553, 298023223876953126, 10314424798490535546559373642, 256923577521058878088611477224235621321608, 6277101735386680763835789423207666416120802188537744130049

Offset: 1

Seiichi Manyama, Jun 16 2019

nonn

Column k=2 of A308676.

Cf. A023887, A062796, A308571, A308671, A308675.

a[n_] := DivisorSum[n, #^(#*n) &]; Array[a, 8] (* Amiram Eldar, May 11 2021 *)

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

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

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

A308676 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(d^k * n/d).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 17, 28, 9, 1, 257, 19684, 273, 6, 1, 65537, 7625597484988, 4294967553, 3126, 24, 1, 4294967297, 443426488243037769948249630619149892804, 340282366920938463463374607431768276993, 298023223876953126, 47450, 8

Offset: 1

Seiichi Manyama, Jun 16 2019

			Square array begins:
   1,    1,          1,                                       1, ...
   3,    5,         17,                                     257, ...
   4,   28,      19684,                           7625597484988, ...
   9,  273, 4294967553, 340282366920938463463374607431768276993, ...

Seiichi Manyama, Antidiagonals n = 1..9, flattened

Columns k=0..3 give A055225, A023887, A308670, A308675.

Cf. A308674.

T[n_, k_] := DivisorSum[n, #^(n * #^(k-1)) &]; Table[T[k, n - k], {n, 1, 7}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

L.g.f. of column k: -log(Product_{j>=1} (1 - j^(j^k) * x^j)^(1/j)).

cp's OEIS Frontend

A308670 a(n) = Sum_{d|n} d^(d*n).

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PARI