A322263 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = numerator of Sum_{d|n} 1/d^k.

1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 7, 2, 1, 17, 28, 21, 6, 4, 1, 33, 82, 73, 26, 2, 2, 1, 65, 244, 273, 126, 25, 8, 4, 1, 129, 730, 1057, 626, 7, 50, 15, 3, 1, 257, 2188, 4161, 3126, 697, 344, 85, 13, 4, 1, 513, 6562, 16513, 15626, 671, 2402, 585, 91, 9, 2, 1, 1025, 19684, 65793, 78126, 23725, 16808, 4369, 757, 13, 12, 6

Offset

1,3

Links

Table of n, a(n) for n=1..78.

Index entries for sequences related to sigma(n)

Formula

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

Dirichlet g.f. of column k: zeta(s)*zeta(s+k) (for rationals Sum_{d|n} 1/d^k).

A(n,k) = numerator of sigma_k(n)/n^k.

Example

Square array begins:
  1,    1,      1,        1,        1,          1,  ...
  2,  3/2,    5/4,      9/8,    17/16,      33/32,  ...
  2,  4/3,   10/9,    28/27,    82/81,    244/243,  ...
  3,  7/4,  21/16,    73/64,  273/256,  1057/1024,  ...
  2,  6/5,  26/25,  126/125,  626/625,  3126/3125,  ...
  4,    2,  25/18,      7/6,  697/648,    671/648,  ...

Mathematica

Table[Function[k, Numerator[DivisorSigma[-k, n]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Numerator[DivisorSigma[k, n]/n^k]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Numerator[SeriesCoefficient[Sum[x^j/(j^k (1 - x^j)), {j, 1, n}], {x, 0, n}]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

Crossrefs

Columns k=0..24 give A000005, A017665, A017667, A017669, A017671, A017673, A017675, A017677, A017679, A017681, A017683, A017685, A017687, A017689, A017691, A017693, A017695, A017697, A017699, A017701, A017703, A017705, A017707, A017709, A017711.

Denominators are in A322264.

Cf. A109974, A279394.

Sequence in context: A058400 A131344 A129262 * A279394 A308509 A280514

Adjacent sequences: A322260 A322261 A322262 * A322264 A322265 A322266

Keyword

nonn,tabl,frac

Author

Ilya Gutkovskiy, Dec 01 2018

Status

approved