A322263 - OEIS

A322263

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

%I #5 Dec 19 2018 13:34:19

%S 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,

%T 244,273,126,25,8,4,1,129,730,1057,626,7,50,15,3,1,257,2188,4161,3126,

%U 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

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

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

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

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

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

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 2, 3/2, 5/4, 9/8, 17/16, 33/32, ...

%e 2, 4/3, 10/9, 28/27, 82/81, 244/243, ...

%e 3, 7/4, 21/16, 73/64, 273/256, 1057/1024, ...

%e 2, 6/5, 26/25, 126/125, 626/625, 3126/3125, ...

%e 4, 2, 25/18, 7/6, 697/648, 671/648, ...

%t Table[Function[k, Numerator[DivisorSigma[-k, n]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

%t Table[Function[k, Numerator[DivisorSigma[k, n]/n^k]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

%t 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

%Y 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.

%Y Denominators are in A322264.

%Y Cf. A109974, A279394.

%K nonn,tabl,frac

%O 1,3

%A _Ilya Gutkovskiy_, Dec 01 2018