A017712 -id:A017712 - cp's OEIS frontend

A017681 Numerator of sum of -9th powers of divisors of n.

1, 513, 19684, 262657, 1953126, 93499, 40353608, 134480385, 387440173, 500976819, 2357947692, 1292535097, 10604499374, 2587675113, 1423901192, 68853957121, 118587876498, 7361363287, 322687697780, 256501107891, 113474345696, 302406791499, 1801152661464, 24510295355

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, 513/512, 19684/19683, 262657/262144, 1953126/1953125, 93499/93312, 40353608/40353607, 134480385/134217728, ...

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

Cf. A017682 (denominator), A013667, A013668.

[Numerator(DivisorSigma(9,n)/n^9): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Table[Numerator[Total[1/Divisors[n]^9]],{n,20}] (* Harvey P. Dale, Aug 26 2013 *)
Table[Numerator[DivisorSigma[9, n]/n^9], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

vector(20, n, numerator(sigma(n, 9)/n^9)) \\ G. C. Greubel, Nov 07 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^9*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017682(n) = zeta(9) (A013667).
Dirichlet g.f. of a(n)/A017682(n): zeta(s)*zeta(s+9).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017682(k) = zeta(10) (A013668). (End)

A017682 Denominator of sum of -9th powers of divisors of n.

Original entry on oeis.org

1, 512, 19683, 262144, 1953125, 93312, 40353607, 134217728, 387420489, 500000000, 2357947691, 1289945088, 10604499373, 2582630848, 1423828125, 68719476736, 118587876497, 7346640384, 322687697779

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, 513/512, 19684/19683, 262657/262144, 1953126/1953125, 93499/93312, 40353608/40353607, 134480385/134217728, ...

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

Cf. A017681.

[Denominator(DivisorSigma(9,n)/n^9): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Table[Denominator[DivisorSigma[9, n]/n^9], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

vector(20, n, denominator(sigma(n, 9)/n^9)) \\ G. C. Greubel, Nov 07 2018

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^9*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

A017683 Numerator of sum of -10th powers of divisors of n.

Original entry on oeis.org

1, 1025, 59050, 1049601, 9765626, 30263125, 282475250, 1074791425, 3486843451, 200195333, 25937424602, 10329823175, 137858491850, 144768565625, 23066408612, 1100586419201, 2015993900450, 3574014537275, 6131066257802, 5125005407613, 16680163512500, 13292930108525

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, 1025/1024, 59050/59049, 1049601/1048576, 9765626/9765625, 30263125/30233088, 282475250/282475249, ...

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

Cf. A017684 (denominator), A013668, A013669.

[Numerator(DivisorSigma(10,n)/n^10): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Table[Numerator[Total[Divisors[n]^-10]],{n,20}] (* Harvey P. Dale, Sep 04 2018 *)
Table[Numerator[DivisorSigma[10, n]/n^10], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

vector(20, n, numerator(sigma(n, 10)/n^10)) \\ G. C. Greubel, Nov 07 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^10*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017684(n) = zeta(10) (A013668).
Dirichlet g.f. of a(n)/A017684(n): zeta(s)*zeta(s+10).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017684(k) = zeta(11) (A013669). (End)

A017684 Denominator of sum of -10th powers of divisors of n.

Original entry on oeis.org

1, 1024, 59049, 1048576, 9765625, 30233088, 282475249, 1073741824, 3486784401, 200000000, 25937424601, 10319560704, 137858491849, 144627327488, 23066015625, 1099511627776, 2015993900449, 3570467226624

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, 1025/1024, 59050/59049, 1049601/1048576, 9765626/9765625, 30263125/30233088, 282475250/282475249, ...

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

Cf. A017683.

[Denominator(DivisorSigma(10,n)/n^10): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Denominator[DivisorSigma[10, n]/n^10], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, denominator(sigma(n, 10)/n^10)) \\ G. C. Greubel, Nov 06 2018

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^10*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

A017685 Numerator of sum of -11th powers of divisors of n.

Original entry on oeis.org

1, 2049, 177148, 4196353, 48828126, 30248021, 1977326744, 8594130945, 31381236757, 50024415087, 285311670612, 185843885311, 1792160394038, 506442812307, 2883268288216, 17600780175361, 34271896307634, 21433384705031, 116490258898220, 102450026512239, 350279478046112

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. A017686 (denominator), A013669, A013670.

[Numerator(DivisorSigma(11,n)/n^11): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Numerator[Total[Divisors[n]^-11]],{n,20}] (* Harvey P. Dale, Aug 26 2012 *)
Table[Numerator[DivisorSigma[11, n]/n^11], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 11)/n^11)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017686(n) = zeta(11) (A013669).
Dirichlet g.f. of a(n)/A017686(n): zeta(s)*zeta(s+11).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017686(k) = zeta(12) (A013670). (End)

A017686 Denominator of sum of -11th powers of divisors of n.

Original entry on oeis.org

1, 2048, 177147, 4194304, 48828125, 30233088, 1977326743, 8589934592, 31381059609, 50000000000, 285311670611, 185752092672, 1792160394037, 506195646208, 2883251953125, 17592186044416, 34271896307633

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A017685.

[Denominator(DivisorSigma(11,n)/n^11): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Denominator[DivisorSigma[-11,Range[20]]] (* Harvey P. Dale, Dec 18 2012 *)

vector(20, n, denominator(sigma(n, 11)/n^11)) \\ G. C. Greubel, Nov 06 2018

A017687 Numerator of sum of -12th powers of divisors of n.

Original entry on oeis.org

1, 4097, 531442, 16781313, 244140626, 1088658937, 13841287202, 68736258049, 282430067923, 500122072361, 3138428376722, 1486382423891, 23298085122482, 28353876833297, 129746582562692, 281543712968705, 582622237229762, 1157115988280531, 2213314919066162, 2048500130460969

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. A017688 (denominator), A013670, A013671.

[Numerator(DivisorSigma(12,n)/n^12): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Numerator[DivisorSigma[12, n]/n^12], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 12)/n^12)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017688(n) = zeta(12) (A013670).
Dirichlet g.f. of a(n)/A017688(n): zeta(s)*zeta(s+12).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017688(k) = zeta(13) (A013671). (End)

A017688 Denominator of sum of -12th powers of divisors of n.

Original entry on oeis.org

1, 4096, 531441, 16777216, 244140625, 1088391168, 13841287201, 68719476736, 282429536481, 500000000000, 3138428376721, 1486016741376, 23298085122481, 28346956187648, 129746337890625, 281474976710656

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. A017687.

[Denominator(DivisorSigma(12,n)/n^12): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Array[Denominator[Total[Divisors[#]^-12]]&,20] (* Harvey P. Dale, Dec 06 2012 *)
Table[Denominator[DivisorSigma[12, n]/n^12], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, denominator(sigma(n, 12)/n^12)) \\ G. C. Greubel, Nov 06 2018

A017689 Numerator of sum of -13th powers of divisors of n.

Original entry on oeis.org

1, 8193, 1594324, 67117057, 1220703126, 1088524711, 96889010408, 549822930945, 2541867422653, 5000610355659, 34522712143932, 26751583696117, 302875106592254, 99226457784093, 648732096885608, 4504149450301441, 9904578032905938, 6941839931265343, 42052983462257060

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. A017690 (denominator), A013671, A013672.

[Numerator(DivisorSigma(13,n)/n^13): n in [1..20]]; // G. C. Greubel, Nov 06 2018

A017689 := proc(n)
    numtheory[sigma][-13](n) ;
    numer(%) ;
end proc: # R. J. Mathar, Sep 21 2017

Table[Numerator[DivisorSigma[13, n]/n^13], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 13)/n^13)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017690(n) = zeta(13) (A013671).
Dirichlet g.f. of a(n)/A017690(n): zeta(s)*zeta(s+13).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017690(k) = zeta(14) (A013672). (End)

A017690 Denominator of sum of -13th powers of divisors of n.

Original entry on oeis.org

1, 8192, 1594323, 67108864, 1220703125, 1088391168, 96889010407, 549755813888, 2541865828329, 5000000000000, 34522712143931, 26748301344768, 302875106592253, 99214346656768, 648731689453125

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. A017689.

[Denominator(DivisorSigma(13,n)/n^13): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Denominator[DivisorSigma[13, n]/n^13], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, denominator(sigma(n, 13)/n^13)) \\ G. C. Greubel, Nov 06 2018

cp's OEIS Frontend

A017681 Numerator of sum of -9th powers of divisors of n.

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A017682 Denominator of sum of -9th powers of divisors of n.

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A017683 Numerator of sum of -10th powers of divisors of n.

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A017684 Denominator of sum of -10th powers of divisors of n.

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A017685 Numerator of sum of -11th powers of divisors of n.

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A017686 Denominator of sum of -11th powers of divisors of n.

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Magma

Mathematica

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A017687 Numerator of sum of -12th powers of divisors of n.

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