A308504 -id:A308504 - cp's OEIS frontend

A294645 a(n) = Sum_{d|n} d^(n+1).

1, 9, 82, 1057, 15626, 282252, 5764802, 134480385, 3486843451, 100048830174, 3138428376722, 107006334784468, 3937376385699290, 155572843119354936, 6568408508343827972, 295150156996346511361, 14063084452067724991010, 708236696816416252145973

Offset: 1

Seiichi Manyama, Nov 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Divisor Function

Column k=1 of A308504.

Cf. A023882, A023887, A082245, A158095, A292312.

Table[DivisorSigma[n + 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Oct 07 2020 *)

PARI
```
{a(n) = sigma(n, n+1)}
```

N=66; x='x+O('x^N); Vec(sum(k=1, N, k^(k+1)*x^k/(1-(k*x)^k)))

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, 1-(k*x)^k)))) \\ Seiichi Manyama, Jun 02 2019

G.f.: Sum_{k>0} k^(k+1)*x^k/(1-(k*x)^k).
L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 02 2019
a(n) ~ n^(n+1). - Vaclav Kotesovec, Oct 07 2020

A308502 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n/d + k).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 9, 1, 17, 28, 25, 6, 1, 33, 82, 81, 26, 24, 1, 65, 244, 289, 126, 80, 8, 1, 129, 730, 1089, 626, 330, 50, 41, 1, 257, 2188, 4225, 3126, 1604, 344, 161, 37, 1, 513, 6562, 16641, 15626, 8634, 2402, 833, 163, 68, 1, 1025, 19684, 66049, 78126, 49100, 16808, 5249, 973, 290, 12

Offset: 1

Seiichi Manyama, Jun 02 2019

			Square array begins:
    1,  1,   1,    1,    1,     1, ...
    3,  5,   9,   17,   33,    65, ...
    4, 10,  28,   82,  244,   730, ...
    9, 25,  81,  289, 1089,  4225, ...
    6, 26, 126,  626, 3126, 15626, ...
   24, 80, 330, 1604, 8634, 49100, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..2 give A055225, A078308, A296601.

Cf. A279394, A308504.

T[n_, k_] := DivisorSum[n, #^(n/# + k) &]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

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

A308570 a(n) = sigma_{2*n}(n).

Original entry on oeis.org

1, 17, 730, 65793, 9765626, 2177317874, 678223072850, 281479271743489, 150094635684419611, 100000095367432689202, 81402749386839761113322, 79496851942053939878082786, 91733330193268616658399616010, 123476696151234472370970011268514

Offset: 1

Seiichi Manyama, Jun 08 2019

nonn

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

Diagonal of A308504.
Column k=2 of A308569.
Cf. A073705.

Table[DivisorSigma[2 n, n], {n, 1, 20}] (* Vaclav Kotesovec, Jun 08 2019 *)

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

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

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

a(n) ~ n^(2*n). - Vaclav Kotesovec, Jun 08 2019

A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 273, 2, 1, 257, 19684, 65793, 3126, 4, 1, 1025, 531442, 16781313, 9765626, 47450, 2, 1, 4097, 14348908, 4295032833, 30517578126, 2177317874, 823544, 4, 1, 16385, 387420490, 1099512676353, 95367431640626, 101560344351050, 678223072850, 16843009, 3

Offset: 1

Seiichi Manyama, Jun 08 2019

			Square array begins:
   1,    1,       1,           1,              1, ...
   2,    5,      17,          65,            257, ...
   2,   28,     730,       19684,         531442, ...
   3,  273,   65793,    16781313,     4295032833, ...
   2, 3126, 9765626, 30517578126, 95367431640626, ...

Seiichi Manyama, Antidiagonals n = 1..52, flattened

Columns k=0..2 give A000005, A023887, A308570.
Rows n=1..2 give A000012, A052539.
A(n,n) gives A308571.
Cf. A308504, A308509.

T[n_, k_] := DivisorSum[n, #^(k*n) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

T(n,k) = sumdiv(n, d, d^(k*n));
matrix(5, 5, n, k, T(n,k-1)) \\ Michel Marcus, Jun 08 2019

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

cp's OEIS Frontend

A294645 a(n) = Sum_{d|n} d^(n+1).

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A308502 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n/d + k).

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A308570 a(n) = sigma_{2*n}(n).

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A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).

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Author

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Examples

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Programs

Mathematica

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Formula