cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 51-60 of 72 results. Next

A017691 Numerator of sum of -14th powers of divisors of n.

Original entry on oeis.org

1, 16385, 4782970, 268451841, 6103515626, 39184481725, 678223072850, 4398314962945, 22876797237931, 10000610353201, 379749833583242, 213999516991295, 3937376385699290, 5556342524323625, 5838586426737844, 72061992352890881, 168377826559400930, 374836322743499435
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017692 (denominator), A013672, A013673.

Programs

  • Magma
    [Numerator(DivisorSigma(14,n)/n^14): n in [1..20]]; // G. C. Greubel, Nov 06 2018
  • Mathematica
    Table[Numerator[DivisorSigma[14, n]/n^14], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)
  • PARI
    vector(20, n, numerator(sigma(n, 14)/n^14)) \\ G. C. Greubel, Nov 06 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017692(n) = zeta(14) (A013672).
Dirichlet g.f. of a(n)/A017692(n): zeta(s)*zeta(s+14).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017692(k) = zeta(15) (A013673). (End)

A017692 Denominator of sum of -14th powers of divisors of n.

Original entry on oeis.org

1, 16384, 4782969, 268435456, 6103515625, 39182082048, 678223072849, 4398046511104, 22876792454961, 10000000000000, 379749833583241, 213986410758144, 3937376385699289, 5556003412779008, 5838585205078125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017691.

Programs

  • Magma
    [Denominator(DivisorSigma(14,n)/n^14): n in [1..20]]; // G. C. Greubel, Nov 06 2018
  • Mathematica
    Table[Denominator[DivisorSigma[14, n]/n^14], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 14)/n^14)) \\ G. C. Greubel, Nov 06 2018

A017693 Numerator of sum of -15th powers of divisors of n.

Original entry on oeis.org

1, 32769, 14348908, 1073774593, 30517578126, 13061093507, 4747561509944, 35185445863425, 205891146443557, 500015258805447, 4177248169415652, 3851873211923611, 51185893014090758, 19446605389919367, 48654880101420712, 1152956690052710401, 2862423051509815794
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017694 (denominator), A013673, A013674.

Programs

  • Magma
    [Numerator(DivisorSigma(15,n)/n^15): n in [1..20]]; // G. C. Greubel, Nov 06 2018
  • Mathematica
    Table[Numerator[DivisorSigma[15, n]/n^15], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)
  • PARI
    vector(20, n, numerator(sigma(n, 15)/n^15)) \\ G. C. Greubel, Nov 06 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017694(n) = zeta(15) (A013673).
Dirichlet g.f. of a(n)/A017694(n): zeta(s)*zeta(s+15).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017694(k) = zeta(16) (A013674). (End)

A017694 Denominator of sum of -15th powers of divisors of n.

Original entry on oeis.org

1, 32768, 14348907, 1073741824, 30517578125, 13060694016, 4747561509943, 35184372088832, 205891132094649, 500000000000000, 4177248169415651, 3851755393646592, 51185893014090757, 19446011944726528, 48654876708984375
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017693.

Programs

  • Magma
    [Denominator(DivisorSigma(15,n)/n^15): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Denominator[DivisorSigma[15, n]/n^15], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 15)/n^15)) \\ G. C. Greubel, Nov 05 2018

A017695 Numerator of sum of -16th powers of divisors of n.

Original entry on oeis.org

1, 65537, 43046722, 4295032833, 152587890626, 1410576509857, 33232930569602, 281479271743489, 1853020231898563, 5000076293978081, 45949729863572162, 30814514057170571, 665416609183179842, 1088993285370003137, 6568408508343827972, 18447025552981295105, 48661191875666868482
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017696 (denominator), A013674, A013675.

Programs

  • Magma
    [Numerator(DivisorSigma(16,n)/n^16): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Numerator[Total[1/Divisors[n]^16]],{n,20}] (* Harvey P. Dale, Sep 26 2014 *)
Table[Numerator[DivisorSigma[16, n]/n^16], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, numerator(sigma(n, 16)/n^16)) \\ G. C. Greubel, Nov 05 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017696(n) = zeta(16) (A013674).
Dirichlet g.f. of a(n)/A017696(n): zeta(s)*zeta(s+16).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017696(k) = zeta(17) (A013675). (End)

A017696 Denominator of sum of -16th powers of divisors of n.

Original entry on oeis.org

1, 65536, 43046721, 4294967296, 152587890625, 1410554953728, 33232930569601, 281474976710656, 1853020188851841, 5000000000000000, 45949729863572161, 30814043149172736, 665416609183179841
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017695.

Programs

  • Magma
    [Denominator(DivisorSigma(16,n)/n^16): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Denominator[DivisorSigma[16, n]/n^16], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 16)/n^16)) \\ G. C. Greubel, Nov 05 2018

A017697 Numerator of sum of -17th powers of divisors of n.

Original entry on oeis.org

1, 131073, 129140164, 17180000257, 762939453126, 1410565726331, 232630513987208, 2251816993685505, 16677181828806733, 50000381469792099, 505447028499293772, 554657012677255537, 8650415919381337934, 3811447419980664273, 32842042032920650888, 295150156996346511361
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017698 (denominator), A013675, A013676.

Programs

  • Magma
    [Numerator(DivisorSigma(17,n)/n^17): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Numerator[DivisorSigma[17, n]/n^17], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, numerator(sigma(n, 17)/n^17)) \\ G. C. Greubel, Nov 05 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017698(n) = zeta(17) (A013675).
Dirichlet g.f. of a(n)/A017698(n): zeta(s)*zeta(s+17).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017698(k) = zeta(18) (A013676). (End)

A017698 Denominator of sum of -17th powers of divisors of n.

Original entry on oeis.org

1, 131072, 129140163, 17179869184, 762939453125, 1410554953728, 232630513987207, 2251799813685248, 16677181699666569, 50000000000000000, 505447028499293771, 554652776685109248, 8650415919381337933
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017697.

Programs

  • Magma
    [Denominator(DivisorSigma(17,n)/n^17): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Denominator[DivisorSigma[17, n]/n^17], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 17)/n^17)) \\ G. C. Greubel, Nov 05 2018

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017700 (denominator), A013676, A013677.

Programs

  • Magma
    [Numerator(DivisorSigma(18,n)/n^18): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Numerator[DivisorSigma[18, n]/n^18], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, numerator(sigma(n, 18)/n^18)) \\ G. C. Greubel, Nov 05 2018

Formula

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)

A017700 Denominator of sum of -18th powers of divisors of n.

Original entry on oeis.org

1, 262144, 387420489, 68719476736, 3814697265625, 50779978334208, 1628413597910449, 18014398509481984, 150094635296999121, 100000000000000000, 5559917313492231481, 4437222213480873984
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Cf. A017699.

Programs

  • Magma
    [Denominator(DivisorSigma(18,n)/n^18): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Denominator[Total[Divisors[n]^-18]],{n,20}] (* Harvey P. Dale, Sep 25 2012 *)
Table[Denominator[DivisorSigma[18, n]/n^18], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 18)/n^18)) \\ G. C. Greubel, Nov 05 2018
Previous Showing 51-60 of 72 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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