A322103 -id:A322103 - 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).

A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

Original entry on oeis.org

1, 1, 5, 1, 7, 7, 1, 11, 13, 17, 1, 19, 31, 35, 11, 1, 35, 85, 95, 31, 35, 1, 67, 247, 311, 131, 91, 15, 1, 131, 733, 1127, 631, 341, 57, 49, 1, 259, 2191, 4295, 3131, 1615, 351, 155, 34, 1, 515, 6565, 16775, 15631, 8645, 2409, 775, 130, 55, 1, 1027, 19687, 66311, 78131, 49111, 16815, 4991, 850, 217, 23

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
   1,   1,    1,     1,     1,      1,  ...
   5,   7,   11,    19,    35,     67,  ...
   7,  13,   31,    85,   247,    733,  ...
  17,  35,   95,   311,  1127,   4295,  ...
  11,  31,  131,   631,  3131,  15631,  ...
  35,  91,  341,  1615,  8645,  49111,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Columns k=0..3 give A060640, A001001, A027847, A027848.

Cf. A109974, A320940 (diagonal), A321876, A322103.

Table[Function[k, Sum[d DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j 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+1)*sigma(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} j*sigma_k(j)*x^j/(1 - x^j).
L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^sigma_k(j)).
A(n,k) = Sum_{d|n} d^(k+1)*sigma_1(n/d).

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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A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

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