A017667 -id:A017667 - cp's OEIS frontend

A017668 Denominator of sum of -2nd powers of divisors of n.

1, 4, 9, 16, 25, 18, 49, 64, 81, 10, 121, 24, 169, 98, 45, 256, 289, 324, 361, 200, 441, 242, 529, 288, 625, 338, 729, 56, 841, 9, 961, 1024, 1089, 578, 49, 432, 1369, 722, 1521, 160, 1681, 441, 1849, 968, 2025, 1058, 2209, 1152, 2401, 500, 2601, 1352, 2809

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, 5/4, 10/9, 21/16, 26/25, 25/18, 50/49, 85/64, 91/81, 13/10, 122/121, 35/24, 170/169, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Colin Defant, On the Density of Ranges of Generalized Divisor Functions, arXiv:1506.05432 [math.NT], 2015.

Cf. A017667 (numerator).

[Denominator(DivisorSigma(2,n)/n^2): n in [1..50]]; // G. C. Greubel, Nov 08 2018

Table[Denominator[DivisorSigma[-2, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[2, n]/n^2], {n, 1, 50}] (* G. C. Greubel, Nov 08 2018 *)

a(n) = denominator(sigma(n, -2)); \\ Michel Marcus, Aug 24 2018

vector(50, n, denominator(sigma(n, 2)/n^2)) \\ G. C. Greubel, Nov 08 2018

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^2*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

A373439 Numerator of sum of reciprocals of square divisors of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 25, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 25, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

			1, 1, 1, 5/4, 1, 1, 1, 5/4, 10/9, 1, 1, 5/4, 1, 1, 1, 21/16, 1, 10/9, 1, 5/4, 1, 1, 1, 5/4, 26/25, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007406, A017665, A017667, A028235, A035316, A332880, A373440 (denominators).

nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Numerator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

a(n) = numerator(sumdiv(n, d, if (issquare(d), 1/d))); \\ Michel Marcus, Jun 05 2024

Numerators of coefficients in expansion of Sum_{k>=1} x^(k^2)/(k^2*(1 - x^(k^2))).
a(n) is the numerator of Sum_{d^2|n} 1/d^2.
From Amiram Eldar, Jun 26 2024: (Start)
Let f(n) = a(n)/A373440(n). Then:
f(n) is multiplicative with f(p^e) = (p^2 - p^(-2*floor(e/2)))/(p^2-1).
Dirichlet g.f. of f(n): zeta(s) * zeta(2*s+2).
Sum_{k=1..n} f(k) ~ zeta(4) * n. (End)

A208767 Generalized 2-super abundant numbers.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 720, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 360360, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 10810800, 12252240, 21621600, 24504480, 36756720, 61261200

Offset: 1

Ben Branman, Mar 01 2012

nonn

The generalized k-super abundant numbers are those such that sigma_k(n)/(n^k) > sigma_k(m)/(m^k) for all m < n, where sigma_k(n) is the sum of the k-th powers of the divisors of n.
1-super abundant numbers are A004394. 0-super abundant numbers are A002182.
Pillai called these numbers "highly abundant numbers of the 2nd order". - Amiram Eldar, Jun 30 2019

			For i=24, sigma_2(24)/(24^2)=850/576=1.47569, a new record, thus 24 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..251
S. Sivasankaranarayana Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
Srinivasa Ramanujan, Highly composite numbers, Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin, The Ramanujan Journal, Vol. 1, No. 2 (1997), pp. 119-153, alternative link.

Cf. A002182, A004394, A004490, A002201, A001157, A013661, A017667, A017668.

Subsequence of A025487.

s = {1}; a = 1; Do[ If[DivisorSigma[2, n]/(n^2) > a, a = DivisorSigma[2, n]/(n^2); AppendTo[s, n]], {n, 10000000}]; s

Limit_{n->oo} A001157(a(n))/a(n)^2 = zeta(2) (A013661). - Amiram Eldar, Sep 25 2022

More terms from Amiram Eldar, May 12 2019

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.

A357555 a(n) is the numerator of Sum_{d|n} (-1)^(d+1) / d^2.

Original entry on oeis.org

1, 3, 10, 11, 26, 5, 50, 43, 91, 39, 122, 55, 170, 75, 52, 171, 290, 91, 362, 143, 500, 183, 530, 215, 651, 255, 820, 275, 842, 13, 962, 683, 1220, 435, 52, 1001, 1370, 543, 1700, 559, 1682, 125, 1850, 61, 2366, 795, 2210, 95, 2451, 1953, 2900, 935, 2810, 205, 3172

Offset: 1

Ilya Gutkovskiy, Oct 03 2022

			1, 3/4, 10/9, 11/16, 26/25, 5/6, 50/49, 43/64, 91/81, 39/50, 122/121, ...

Cf. A017667, A064027, A098987, A119682, A321543, A357556 (denominators).

Table[Sum[(-1)^(d + 1)/d^2, {d, Divisors[n]}], {n, 1, 55}] // Numerator
nmax = 55; CoefficientList[Series[Sum[(-1)^(k + 1) x^k/(k^2 (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest

a(n) = numerator(sumdiv(n, d, (-1)^(d+1)/d^2)); \\ Michel Marcus, Oct 03 2022

from sympy import divisors
from fractions import Fraction
def a(n): return sum(Fraction((-1)**(d+1), d*d) for d in divisors(n, generator=True)).numerator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022

Numerators of coefficients in expansion of Sum_{k>=1} (-1)^(k+1) * x^k / (k^2 * (1 - x^k)).

A384817 Numerator of the sum of the reciprocals of all square divisors of all positive integers <= n.

Original entry on oeis.org

1, 2, 3, 17, 21, 25, 29, 17, 173, 191, 209, 463, 499, 535, 571, 2473, 2617, 2777, 2921, 3101, 3245, 3389, 3533, 3713, 96569, 100169, 34723, 36223, 37423, 38623, 39823, 20699, 21299, 21899, 22499, 69997, 71797, 73597, 75397, 77647, 79447, 81247, 83047, 85297

Offset: 1

Ilya Gutkovskiy, Jun 10 2025

			1, 2, 3, 17/4, 21/4, 25/4, 29/4, 17/2, 173/18, 191/18, 209/18, 463/36, ...

Cf. A007406, A017667, A284648, A309125, A373439, A384818 (denominators).

nmax = 44; CoefficientList[Series[1/(1 - x) Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
Table[Sum[Floor[n/k^2]/k^2, {k, 1, Floor[Sqrt[n]]}], {n, 1, 44}] // Numerator

a(n) = numerator(sum(k=1, n, sumdiv(k, d, if (issquare(d), 1/d)))); \\ Michel Marcus, Jun 10 2025

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^(k^2) / (k^2*(1 - x^(k^2))).
a(n) is the numerator of Sum_{k=1..floor(sqrt(n))} floor(n/k^2) / k^2.
a(n) / A384818(n) ~ Pi^4 * n / 90.

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A017668 Denominator of sum of -2nd powers of divisors of n.

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A373439 Numerator of sum of reciprocals of square divisors of n.

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A208767 Generalized 2-super abundant numbers.

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

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A357555 a(n) is the numerator of Sum_{d|n} (-1)^(d+1) / d^2.

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A384817 Numerator of the sum of the reciprocals of all square divisors of all positive integers <= n.

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