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

1,3

Links

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

Formula

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).

Example

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, ...

Prog

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

Crossrefs

Columns k=0..3 give A061201, A280077, A356042, A356043.

T(n,n) gives A356046.

Cf. A322103.

Sequence in context: A058662 A344026 A133866 * A242131 A177266 A356124

Adjacent sequences: A356042 A356043 A356044 * A356046 A356047 A356048

Keyword

nonn,tabl

Author

Seiichi Manyama, Jul 24 2022

Status

approved