A321140 -id:A321140 - cp's OEIS frontend

A321141 a(n) = Sum_{d|n} sigma_n(d).

1, 6, 29, 291, 3127, 48246, 823545, 16909060, 387459858, 10019533302, 285311670613, 8920489178073, 302875106592255, 11113363271736486, 437893951444713443, 18447307036548136965, 827240261886336764179, 39346708467688595378892, 1978419655660313589123981

Offset: 1

Ilya Gutkovskiy, Oct 28 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386
N. J. A. Sloane, Transforms

Cf. A000005, A007429, A007433, A023887, A319194, A320940, A321140.

[&+[DivisorSigma(n, d):d in Divisors(n)]:n in [1..20]]; // Vincenzo Librandi, Feb 16 2020

with(numtheory): seq(coeff(series(add(sigma[n](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018

Table[Sum[DivisorSigma[n, d], {d, Divisors[n]}] , {n, 19}]
Table[SeriesCoefficient[Sum[DivisorSigma[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 19}]

a(n) = sumdiv(n, d, sigma(d, n)); \\ Michel Marcus, Oct 28 2018

from sympy import divisor_sigma, divisors
def A321141(n):
    return sum(divisor_sigma(d,0)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = [x^n] Sum_{k>=1} sigma_n(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d^n*tau(n/d).
a(n) ~ n^n. - Vaclav Kotesovec, Feb 16 2020

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).

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

A321141 a(n) = Sum_{d|n} sigma_n(d).

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

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