A308674 -id:A308674 - cp's OEIS frontend

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).

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)).

A308671 a(n) = Sum_{d|n} d^(d^2).

Original entry on oeis.org

1, 17, 19684, 4294967313, 298023223876953126, 10314424798490535546171968756, 256923577521058878088611477224235621321608, 6277101735386680763835789423207666416102355444468329480209

Offset: 1

Seiichi Manyama, Jun 16 2019

nonn

Column k=2 of A308674.

Cf. A000005, A062796, A308670.

a[n_] := DivisorSum[n, #^(#^2) &]; Array[a, 8] (* Amiram Eldar, May 11 2021 *)

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

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

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

A308672 a(n) = Sum_{d|n} d^(d^3).

Original entry on oeis.org

1, 257, 7625597484988, 340282366920938463463374607431768211713, 2350988701644575015937473074444491355637331113544175043017503412556834518909454345703126

Offset: 1

Seiichi Manyama, Jun 16 2019

nonn

The next term (a(6)) has 169 digits. - Harvey P. Dale, Sep 08 2020

Column k=3 of A308674.

Cf. A000005, A062796, A308671.

Table[Total[Divisors[n]^Divisors[n]^3],{n,5}] (* Harvey P. Dale, Sep 08 2020 *)
a[n_] := DivisorSum[n, #^(#^3) &]; Array[a, 5] (* Amiram Eldar, May 11 2021 *)

PARI
```
{a(n) = sumdiv(n, d, d^d^3)}
```

N=10; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^(k^(k^3-1))))))

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

cp's OEIS Frontend

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).

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A308671 a(n) = Sum_{d|n} d^(d^2).

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Mathematica

PARI

PARI

Formula

A308672 a(n) = Sum_{d|n} d^(d^3).

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Mathematica

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