A017712 -id:A017712 - 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

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

A013962 a(n) = sigma_14(n), the sum of the 14th powers of the divisors of n.

Original entry on oeis.org

1, 16385, 4782970, 268451841, 6103515626, 78368963450, 678223072850, 4398314962945, 22876797237931, 100006103532010, 379749833583242, 1283997101947770, 3937376385699290, 11112685048647250, 29192932133689220, 72061992352890881, 168377826559400930

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013673, A013954-A013972, A017665-A017712.

[DivisorSigma(14, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[14,Range[20]] (* Harvey P. Dale, Mar 10 2013 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^14*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 14); \\ Amiram Eldar, Oct 29 2023

[sigma(n,14) for n in range(1,16)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^14*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-14)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(14*e+14)-1)/(p^14-1).
Sum_{k=1..n} a(k) = zeta(15) * n^15 / 15 + O(n^16). (End)

A013966 a(n) = sigma_18(n), the sum of the 18th powers of the divisors of n.

Original entry on oeis.org

1, 262145, 387420490, 68719738881, 3814697265626, 101560344351050, 1628413597910450, 18014467229220865, 150094635684419611, 1000003814697527770, 5559917313492231482, 26623434909949071690, 112455406951957393130, 426880482624234915250, 1477891883850485076740

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013677, A013954-A013972, A017665-A017712.

[DivisorSigma(18,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Table[DivisorSigma[18,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

a(n)=sigma(n,18) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n,18)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^18*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(18*e+18)-1)/(p^18-1).
Dirichlet g.f.: zeta(s)*zeta(s-18).
Sum_{k=1..n} a(k) = zeta(19) * n^19 / 19. + O(n^20). (End)

A013971 a(n) = sigma_23(n), the sum of the 23rd powers of the divisors of n.

Original entry on oeis.org

1, 8388609, 94143178828, 70368752566273, 11920928955078126, 789730317205170252, 27368747340080916344, 590295880727458217985, 8862938119746644274757, 100000011920928963466734, 895430243255237372246532, 6624738056749922960468044, 41753905413413116367045798

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(23,n): n in [1..30]]; // G. C. Greubel, Nov 03 2018

DivisorSigma[23,Range[15]] (* Harvey P. Dale, May 02 2016 *)

vector(30, n, sigma(n,23)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,23)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^23*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(23*e+23)-1)/(p^23-1).
Dirichlet g.f.: zeta(s)*zeta(s-23).
Sum_{k=1..n} a(k) = zeta(24) * n^24 / 24 + O(n^25). (End)

A013964 a(n) = sigma_16(n), the sum of the 16th powers of the divisors of n.

Original entry on oeis.org

1, 65537, 43046722, 4295032833, 152587890626, 2821153019714, 33232930569602, 281479271743489, 1853020231898563, 10000152587956162, 45949729863572162, 184887084343023426, 665416609183179842, 2177986570740006274, 6568408508343827972, 18447025552981295105

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013675, A013954-A013972, A017665-A017712.

[DivisorSigma(16, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[16, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^16*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 16); \\ Amiram Eldar, Oct 29 2023

[sigma(n,16)for n in range(1,14)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^16*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-16)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(16*e+16)-1)/(p^16-1).
Sum_{k=1..n} a(k) = zeta(17) * n^17 / 17 + O(n^18). (End)

A013968 a(n) = sigma_20(n), the sum of the 20th powers of the divisors of n.

Original entry on oeis.org

1, 1048577, 3486784402, 1099512676353, 95367431640626, 3656161927895954, 79792266297612002, 1152922604119523329, 12157665462543713203, 100000095367432689202, 672749994932560009202, 3833763649708914645906, 19004963774880799438802, 83668335217551100221154

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(20,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

DivisorSigma[20,Range[20]] (* Harvey P. Dale, Jul 26 2015 *)

vector(50, n, sigma(n,20)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,20)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^20*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(20*e+20)-1)/(p^20-1).
Dirichlet g.f.: zeta(s)*zeta(s-20).
Sum_{k=1..n} a(k) = zeta(21) * n^21 / 21 + O(n^22). (End)

A013970 a(n) = sigma_22(n), the sum of the 22nd powers of the divisors of n.

Original entry on oeis.org

1, 4194305, 31381059610, 17592190238721, 2384185791015626, 131621735227521050, 3909821048582988050, 73786993887028445185, 984770902214992292491, 10000002384185795209930, 81402749386839761113322, 552061570551763831158810, 3211838877954855105157370

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(22,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

lst={};Do[AppendTo[lst,DivisorSigma[22,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
a[ n_] := DivisorSigma[ 22, n]; (* Michael Somos, Dec 19 2016 *)

vector(50, n, sigma(n,22)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,22)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^22*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(22*e+22)-1)/(p^22-1).
Dirichlet g.f.: zeta(s)*zeta(s-22).
Sum_{k=1..n} a(k) = zeta(23) * n^23 / 23 + O(n^24). (End)

A017669 Numerator of sum of -3rd powers of divisors of n.

Original entry on oeis.org

1, 9, 28, 73, 126, 7, 344, 585, 757, 567, 1332, 511, 2198, 387, 392, 4681, 4914, 757, 6860, 4599, 1376, 2997, 12168, 455, 15751, 9891, 20440, 3139, 24390, 147, 29792, 37449, 4144, 22113, 6192, 55261, 50654, 15435, 61544, 7371, 68922, 172, 79508, 24309, 10598

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, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A017670 (denominator), A002117, A013662.

[Numerator(DivisorSigma(3,n)/n^3): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Numerator[DivisorSigma[-3, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, numerator(sigma(n, 3)/n^3)) \\ G. C. Greubel, Nov 08 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017670(n) = zeta(3) (A002117).
Dirichlet g.f. of a(n)/A017670(n): zeta(s)*zeta(s+3).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017670(k) = zeta(4) (A013662). (End)

A017670 Denominator of sum of -3rd powers of divisors of n.

Original entry on oeis.org

1, 8, 27, 64, 125, 6, 343, 512, 729, 500, 1331, 432, 2197, 343, 375, 4096, 4913, 648, 6859, 4000, 1323, 2662, 12167, 384, 15625, 8788, 19683, 2744, 24389, 125, 29791, 32768, 3993, 19652, 6125, 46656, 50653, 13718, 59319, 6400, 68921, 147, 79507, 21296, 10125

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, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...

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

Cf. A017669.

[Denominator(DivisorSigma(3,n)/n^3): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Denominator[DivisorSigma[-3, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, denominator(sigma(n, 3)/n^3)) \\ G. C. Greubel, Nov 08 2018

Denominator of Sum_{d|n} 1/d^3.

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

cp's OEIS Frontend

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

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

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A013962 a(n) = sigma_14(n), the sum of the 14th powers of the divisors of n.

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A013966 a(n) = sigma_18(n), the sum of the 18th powers of the divisors of n.

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A013971 a(n) = sigma_23(n), the sum of the 23rd powers of the divisors of n.

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A013964 a(n) = sigma_16(n), the sum of the 16th powers of the divisors of n.

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