A356043 -id:A356043 - cp's OEIS frontend

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).

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).

A356126 a(n) = Sum_{k=1..n} k * sigma_3(k).

Original entry on oeis.org

1, 19, 103, 395, 1025, 2537, 4945, 9625, 16438, 27778, 42430, 66958, 95532, 138876, 191796, 266692, 350230, 472864, 603204, 787164, 989436, 1253172, 1533036, 1926156, 2319931, 2834263, 3386143, 4089279, 4796589, 5749149, 6672701, 7871069, 9101837, 10605521

Offset: 1

Seiichi Manyama, Jul 27 2022

nonn

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

Partial sums of A281372.
Column k=4 of A356124.
Cf. A356043.

a[n_] := Sum[k * DivisorSigma[3, k], {k, 1, n}]; Array[a, 34] (* Amiram Eldar, Jul 28 2022 *)

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

a(n) = sum(k=1, n, k^4*binomial(n\k+1, 2));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k/(1-x^k)^2)/(1-x))

from math import isqrt
def A356126(n): return ((-(s:=isqrt(n))**2*(s+1)**2*((s<<1)+1)*(s*(3*(s+1))-1)>>1)+sum(k*(q:=n//k)*(q+1)*(15*k**3+((q<<1)+1)*(q*(3*(q+1))-1)) for k in range(1,s+1)))//30 # Chai Wah Wu, Oct 24 2023

a(n) = Sum_{k=1..n} k^4 * binomial(floor(n/k)+1,2).

G.f.: (1/(1-x)) * Sum_{k>=1} k^4 * x^k/(1 - x^k)^2.

cp's OEIS Frontend

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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A356126 a(n) = Sum_{k=1..n} k * sigma_3(k).

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