A356046 -id:A356046 - 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).

A356128 a(n) = Sum_{k=1..n} k * sigma_n(k).

Original entry on oeis.org

1, 11, 103, 1373, 20657, 381795, 7921825, 187452793, 4916743582, 142471278944, 4506381463150, 154747691135574, 5729252807696052, 227595085199164036, 9654855890695727316, 435664037303036699736, 20836069678062430493950, 1052867409176853099312712

Offset: 1

Seiichi Manyama, Jul 27 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A356046, A356124.

a[n_] := Sum[k * DivisorSigma[n, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 28 2022 *)

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

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

from math import isqrt
from sympy import bernoulli
def A356128(n): return ((s:=isqrt(n))*(s+1)*(bernoulli(n+2)-bernoulli(n+2,s+1))+sum(k**(n+1)*(n+2)*(q:=n//k)*(q+1)+(k*(bernoulli(n+2,q+1)-bernoulli(n+2))<<1) for k in range(1,s+1)))//(n+2)>>1 # Chai Wah Wu, Oct 24 2023

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

a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} k^(n+1) * x^k/(1 - x^k)^2.

A356076 a(n) = Sum_{k=1..n} sigma_k(k) * floor(n/k).

Original entry on oeis.org

1, 7, 36, 315, 3442, 50926, 874471, 17717759, 405157961, 10414927743, 295726598356, 9214021189459, 312089127781714, 11424774177252514, 449318695090042077, 18896344248088180470, 846136606134424944649, 40192694877626991357901

Offset: 1

Seiichi Manyama, Jul 25 2022

nonn

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

Partial sums of A344434.

Cf. A023887, A356046.

Table[Sum[DivisorSigma[k,k]Floor[n/k],{k,n}],{n,20}] (* Harvey P. Dale, Sep 08 2024 *)

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k))/(1-x))

from sympy import divisor_sigma
def A356079(n): return n+sum(divisor_sigma(k,k)*(n//k) for k in range(2,n+1)) # Chai Wah Wu, Jul 25 2022

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

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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A356128 a(n) = Sum_{k=1..n} k * sigma_n(k).

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A356076 a(n) = Sum_{k=1..n} sigma_k(k) * floor(n/k).

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Mathematica

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