A017668 -id:A017668 - cp's OEIS frontend

A261989 Numbers such that A017666(n) = A017668(n).

1, 65, 175, 20737, 32045, 41474, 64090, 128180, 207370, 207553, 352529, 415106, 705058, 1264957, 1762645, 1824877, 2075530, 2529914, 3525290, 3628975, 3649754, 7050580, 7590250, 10223341, 12649570, 17626450, 20446682, 21504269, 23723401, 36321775, 40893364, 43008538, 45621925

Offset: 1

Michel Marcus, Sep 19 2015

nonn

			sigma(65,-1) = 84/65 and sigma(65,-2) = 68/65, both having the same denominator, so 65 is a term.
sigma(32045, -1) = 9072/6409 and sigma(32045, -2) = 6736/6409, both having the same denominator, so 32045 is a term.

Michel Marcus, Table of n, a(n) for n = 1..61

Cf. denominator of sum of powers of divisors of n: A017666 (-1st powers), A017668 (-2nd powers).

isok(n) = denominator(sigma(n,-1))==denominator(sigma(n,-2));

A017667 Numerator of sum of -2nd powers of divisors of n.

Original entry on oeis.org

1, 5, 10, 21, 26, 25, 50, 85, 91, 13, 122, 35, 170, 125, 52, 341, 290, 455, 362, 273, 500, 305, 530, 425, 651, 425, 820, 75, 842, 13, 962, 1365, 1220, 725, 52, 637, 1370, 905, 1700, 221, 1682, 625, 1850, 1281, 2366, 1325, 2210, 1705, 2451, 651, 2900, 1785

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
Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^2*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018
C. Defant proves that there are no positive integers n such that sigma_{-2}(n) lies in (Pi^2/8, 5/4). See arxiv link. - Michel Marcus, Aug 24 2018

			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, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 3 (2015), pp. 80-87; arXiv preprint, arXiv:1506.05432 [math.NT], 2015.

Cf. A017668 (denominator), A002117, A013661, A111003 (Pi^2/8).

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

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

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

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

Dirichlet g.f.: zeta(s)*zeta(s+2) [for fraction A017667/A017668]. - Franklin T. Adams-Watters, Sep 11 2005
sup_{n>=1} a(n)/A017668(n) = zeta(2) (A013661). - Amiram Eldar, Sep 25 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017668(k) = zeta(3) (A002117). - Amiram Eldar, Apr 02 2024

A373440 Denominator of sum of reciprocals of square divisors of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 18, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 18, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 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. A007407, A007947, A017666, A017668, A035316, A332881, A373439 (numerators).

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

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

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

a(n) is the denominator of Sum_{d^2|n} 1/d^2.

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

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.

A357556 a(n) is the denominator of Sum_{d|n} (-1)^(d+1) / d^2.

Original entry on oeis.org

1, 4, 9, 16, 25, 6, 49, 64, 81, 50, 121, 72, 169, 98, 45, 256, 289, 108, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 15, 961, 1024, 1089, 578, 49, 1296, 1369, 722, 1521, 800, 1681, 147, 1849, 88, 2025, 1058, 2209, 128, 2401, 2500, 2601, 1352, 2809, 243, 3025

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. A017668, A064027, A098988, A321543, A334580, A357555 (numerators).

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

a(n) = denominator(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)).denominator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022

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

A384818 Denominator of the sum of the reciprocals of all square divisors of all positive integers <= n.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 2, 18, 18, 18, 36, 36, 36, 36, 144, 144, 144, 144, 144, 144, 144, 144, 144, 3600, 3600, 1200, 1200, 1200, 1200, 1200, 600, 600, 600, 600, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 600, 600, 600, 1200, 58800, 58800, 58800, 58800, 58800

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. A007407, A017668, A284650, A309125, A373440, A384817 (numerators).

nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[Floor[n/k^2]/k^2, {k, 1, Floor[Sqrt[n]]}], {n, 1, 53}] // Denominator

a(n) = denominator(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 denominator of Sum_{k=1..floor(sqrt(n))} floor(n/k^2) / k^2.
A384817(n) / a(n) ~ Pi^4 * n / 90.

cp's OEIS Frontend

A261989 Numbers such that A017666(n) = A017668(n).

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A017667 Numerator of sum of -2nd powers of divisors of n.

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A373440 Denominator of sum of reciprocals of square divisors of n.

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A208767 Generalized 2-super abundant numbers.

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

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A357556 a(n) is the denominator of Sum_{d|n} (-1)^(d+1) / d^2.

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A384818 Denominator of the sum of the reciprocals of all square divisors of all positive integers <= n.

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