A356042 -id:A356042 - cp's OEIS frontend

A356125 a(n) = Sum_{k=1..n} k * sigma_2(k).

1, 11, 41, 125, 255, 555, 905, 1585, 2404, 3704, 5046, 7566, 9776, 13276, 17176, 22632, 27562, 35752, 42630, 53550, 64050, 77470, 89660, 110060, 126335, 148435, 170575, 199975, 224393, 263393, 293215, 336895, 377155, 426455, 471955, 540751, 591441, 660221, 726521

Offset: 1

Seiichi Manyama, Jul 27 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Partial sums of A328259.
Column k=3 of A356124.
Cf. A319086, A356042.

a[n_] := Sum[k * DivisorSigma[2, k], {k, 1, n}]; Array[a, 39] (* Amiram Eldar, Jul 28 2022 *)

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

a(n) = sum(k=1, n, k^3*binomial(n\k+1, 2));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^3*x^k/(1-x^k)^2)/(1-x))

from math import isqrt
def A356125(n): return (-((s:=isqrt(n))*(s+1))**3>>1) + sum(k*(q:=n//k)*(q+1)*(2*k**2+q*(q+1)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^3 * binomial(floor(n/k)+1,2).
G.f.: (1/(1-x)) * Sum_{k>=1} k^3 * x^k/(1 - x^k)^2.
a(n) ~ zeta(3) * n^4 / 4. - Vaclav Kotesovec, Aug 02 2022

A356045 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} sigma_k(j) * floor(n/j).

Original entry on oeis.org

1, 1, 4, 1, 5, 7, 1, 7, 10, 13, 1, 11, 18, 21, 16, 1, 19, 40, 45, 28, 25, 1, 35, 102, 123, 72, 48, 28, 1, 67, 280, 393, 250, 138, 57, 38, 1, 131, 798, 1371, 1020, 540, 189, 83, 44, 1, 259, 2320, 5025, 4498, 2514, 885, 301, 101, 53, 1, 515, 6822, 18963, 20652, 12828, 4917, 1553, 403, 129, 56

Offset: 1

Seiichi Manyama, Jul 24 2022

			Square array begins:
   1,  1,   1,   1,    1,     1, ...
   4,  5,   7,  11,   19,    35, ...
   7, 10,  18,  40,  102,   280, ...
  13, 21,  45, 123,  393,  1371, ...
  16, 28,  72, 250, 1020,  4498, ...
  25, 48, 138, 540, 2514, 12828, ...

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

Columns k=0..3 give A061201, A280077, A356042, A356043.
T(n,n) gives A356046.
Cf. A322103.

T(n, k) = sum(j=1, n, sigma(j, k)*(n\j));

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

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

G.f. of column k: (1/(1-x)) * Sum_{j>=1} sigma_k(j) * x^j/(1 - x^j).
T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k * tau(j/d).
T(n,k) = Sum_{j=1..n} Sum_{d|j} sigma_k(d).

cp's OEIS Frontend

A356125 a(n) = Sum_{k=1..n} k * sigma_2(k).

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A356045 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} sigma_k(j) * floor(n/j).

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