A356045 -id:A356045 - cp's OEIS frontend

A356046 a(n) = Sum_{k=1..n} sigma_n(k) * floor(n/k).

1, 7, 40, 393, 4498, 68898, 1205205, 24830617, 574911611, 14936215765, 427782762142, 13426870089265, 457622727372932, 16842615801316402, 665489035541044561, 28102162770144986248, 1262904298391426474369, 60182778141796948356895

Offset: 1

Seiichi Manyama, Jul 24 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A321141, A356045.

PARI
```
a(n) = sum(k=1, n, sigma(k, n)*(n\k));
```

a(n) = sum(k=1, n, sumdiv(k, d, d^n*numdiv(k/d)));

a(n) = sum(k=1, n, sumdiv(k, d, sigma(d, n)));

a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} sigma_n(k) * x^k/(1 - x^k).
a(n) = Sum_{k=1..n} Sum_{d|k} d^n * tau(k/d).
a(n) = Sum_{k=1..n} Sum_{d|k} sigma_n(d).
a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Aug 07 2022

A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 1, 6, 5, 6, 1, 10, 11, 11, 3, 1, 18, 29, 27, 7, 9, 1, 34, 83, 83, 27, 20, 3, 1, 66, 245, 291, 127, 66, 9, 10, 1, 130, 731, 1091, 627, 290, 51, 26, 6, 1, 258, 2189, 4227, 3127, 1494, 345, 112, 18, 9, 1, 514, 6563, 16643, 15627, 8330, 2403, 668, 102, 28, 3

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  1,   1,   1,    1,     1,     1,  ...
  3,   4,   6,   10,    18,    34,  ...
  3,   5,  11,   29,    83,   245,  ...
  6,  11,  27,   83,   291,  1091,  ...
  3,   7,  27,  127,   627,  3127,  ...
  9,  20,  66,  290,  1494,  8330,  ...

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

Columns k=0..3 give A007425, A007429, A007433, A321140.

Cf. A109974, A321141 (diagonal), A356045.

Table[Function[k, Sum[DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, d^k*numdiv(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} sigma_k(j)*x^j/(1 - x^j).

A(n,k) = Sum_{d|n} d^k*tau(n/d).

A356042 a(n) = Sum_{k=1..n} sigma_2(k) * floor(n/k).

Original entry on oeis.org

1, 7, 18, 45, 72, 138, 189, 301, 403, 565, 688, 985, 1156, 1462, 1759, 2212, 2503, 3115, 3478, 4207, 4768, 5506, 6037, 7269, 7947, 8973, 9895, 11272, 12115, 13897, 14860, 16678, 18031, 19777, 21154, 23908, 25279, 27457, 29338, 32362, 34045, 37411, 39262, 42583

Offset: 1

Seiichi Manyama, Jul 24 2022

nonn

Partial sums of A007433.
Column k=2 of A356045.
Cf. A000005 (tau).

Table[Sum[DivisorSigma[2, k]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 07 2022 *)

PARI
```
a(n) = sum(k=1, n, sigma(k, 2)*(n\k));
```

a(n) = sum(k=1, n, sumdiv(k, d, d^2*numdiv(k/d)));

my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, 2)*x^k/(1-x^k))/(1-x))

a(n) = Sum_{k=1..n} Sum_{d|k} d^2 * tau(k/d).
G.f.: (1/(1-x)) * Sum_{k>=1} sigma_2(k) * x^k/(1 - x^k).
a(n) ~ zeta(3)^2 * n^3 / 3. - Vaclav Kotesovec, Aug 07 2022

A356043 a(n) = Sum_{k=1..n} sigma_3(k) * floor(n/k).

Original entry on oeis.org

1, 11, 40, 123, 250, 540, 885, 1553, 2339, 3609, 4942, 7349, 9548, 12998, 16681, 22030, 26945, 34805, 41666, 52207, 62212, 75542, 87711, 107083, 122961, 144951, 166177, 194812, 219203, 256033, 285826, 328624, 367281, 416431, 460246, 525484, 576139, 644749, 708520

Offset: 1

Seiichi Manyama, Jul 24 2022

nonn

Partial sums of A321140.
Column k=3 of A356045.
Cf. A000005 (tau).

Table[Sum[DivisorSigma[3, k]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 07 2022 *)

PARI
```
a(n) = sum(k=1, n, sigma(k, 3)*(n\k));
```

a(n) = sum(k=1, n, sumdiv(k, d, d^3*numdiv(k/d)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, 3)*x^k/(1-x^k))/(1-x))

a(n) = Sum_{k=1..n} Sum_{d|k} d^3 * tau(k/d).
G.f.: (1/(1-x)) * Sum_{k>=1} sigma_3(k) * x^k/(1 - x^k).
a(n) ~ Pi^8 * n^4 / 32400. - Vaclav Kotesovec, Aug 07 2022

cp's OEIS Frontend

A356046 a(n) = Sum_{k=1..n} sigma_n(k) * floor(n/k).

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A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).

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A356042 a(n) = Sum_{k=1..n} sigma_2(k) * floor(n/k).

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A356043 a(n) = Sum_{k=1..n} sigma_3(k) * floor(n/k).

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Mathematica

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