A017700 -id:A017700 - cp's OEIS frontend

A017699 Numerator of sum of -18th powers of divisors of n.

1, 262145, 387420490, 68719738881, 3814697265626, 50780172175525, 1628413597910450, 18014467229220865, 150094635684419611, 100000381469752777, 5559917313492231482, 4437239151658178615, 112455406951957393130, 213440241312117457625, 295578376770097015348

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

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

Cf. A017700 (denominator), A013676, A013677.

[Numerator(DivisorSigma(18,n)/n^18): n in [1..20]]; // G. C. Greubel, Nov 05 2018

Table[Numerator[DivisorSigma[18, n]/n^18], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)

vector(20, n, numerator(sigma(n, 18)/n^18)) \\ G. C. Greubel, Nov 05 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017700(n) = zeta(18) (A013676).
Dirichlet g.f. of a(n)/A017700(n): zeta(s)*zeta(s+18).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017700(k) = zeta(19) (A013677). (End)

A322264 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{d|n} 1/d^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 4, 1, 1, 16, 27, 16, 5, 1, 1, 32, 81, 64, 25, 1, 1, 1, 64, 243, 256, 125, 18, 7, 1, 1, 128, 729, 1024, 625, 6, 49, 8, 1, 1, 256, 2187, 4096, 3125, 648, 343, 64, 9, 1, 1, 512, 6561, 16384, 15625, 648, 2401, 512, 81, 5, 1, 1, 1024, 19683, 65536, 78125, 23328, 16807, 4096, 729, 10, 11, 1

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 A000012, A017666, A017668, A017670, A017672, A017674, A017676, A017678, A017680, A017682, A017684, A017686, A017688, A017690, A017692, A017694, A017696, A017698, A017700, A017702, A017704, A017706, A017708, A017710, A017712.
Numerators are in A322263.
Cf. A109974, A279394.

Table[Function[k, Denominator[DivisorSigma[-k, n]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[DivisorSigma[k, n]/n^k]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[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) = denominator of sigma_k(n)/n^k.

cp's OEIS Frontend

A017699 Numerator of sum of -18th powers of divisors of n.

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