A017679 -id:A017679 - cp's OEIS frontend

A017680 Denominator of sum of -8th powers of divisors of n.

1, 256, 6561, 65536, 390625, 839808, 5764801, 16777216, 43046721, 50000000, 214358881, 71663616, 815730721, 737894528, 2562890625, 4294967296, 6975757441, 11019960576, 16983563041, 12800000000

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 257/256, 6562/6561, 65793/65536, 390626/390625, 843217/839808, 5764802/5764801, 16843009/16777216, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017679.

[Denominator(DivisorSigma(8,n)/n^8): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Table[Denominator[Total[1/Divisors[n]^8]],{n,20}] (* Harvey P. Dale, Dec 16 2013 *)
Table[Denominator[DivisorSigma[8, n]/n^8], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

vector(20, n, denominator(sigma(n, 8)/n^8)) \\ G. C. Greubel, Nov 07 2018

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^8*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Dec 01 2018

			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,  ...

Index entries for sequences related to sigma(n)

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.

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

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.

cp's OEIS Frontend

A017680 Denominator of sum of -8th powers of divisors of n.

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