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 81-90 of 97 results. Next

A017702 Denominator of sum of -19th powers of divisors of n.

Original entry on oeis.org

1, 524288, 1162261467, 274877906944, 19073486328125, 50779978334208, 11398895185373143, 144115188075855872, 1350851717672992089, 5000000000000000000, 61159090448414546291, 79869999842655731712
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. A017701.

Programs

  • Magma
    [Denominator(DivisorSigma(19,n)/n^19): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Denominator[DivisorSigma[19, n]/n^19], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 19)/n^19)) \\ G. C. Greubel, Nov 05 2018

A017703 Numerator of sum of -20th powers of divisors of n.

Original entry on oeis.org

1, 1048577, 3486784402, 1099512676353, 95367431640626, 1828080963947977, 79792266297612002, 1152922604119523329, 12157665462543713203, 50000047683716344601, 672749994932560009202, 638960608284819107651, 19004963774880799438802, 41834167608775550110577
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. A017704 (denominator), A013678.

Programs

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

Formula

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017704(n) = zeta(20) (A013678).
Dirichlet g.f. of a(n)/A017704(n): zeta(s)*zeta(s+20).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017704(k) = zeta(21). (End)

A017704 Denominator of sum of -20th powers of divisors of n.

Original entry on oeis.org

1, 1048576, 3486784401, 1099511627776, 95367431640625, 1828079220031488, 79792266297612001, 1152921504606846976, 12157665459056928801, 50000000000000000000, 672749994932560009201, 638959998741245853696
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. A017703.

Programs

  • Magma
    [Denominator(DivisorSigma(20,n)/n^20): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Denominator[DivisorSigma[-20,Range[20]]] (* Harvey P. Dale, Dec 31 2014 *)
Table[Denominator[DivisorSigma[20, n]/n^20], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 20)/n^20)) \\ G. C. Greubel, Nov 05 2018

A017705 Numerator of sum of -21st powers of divisors of n.

Original entry on oeis.org

1, 2097153, 10460353204, 4398048608257, 476837158203126, 609360030634117, 558545864083284008, 9223376434903384065, 109418989141972712413, 500000238418580150139, 7400249944258160101212, 11501285462682212701357, 247064529073450392704414, 146419516812481413403653
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. A017706 (denominator).

Programs

  • Magma
    [Numerator(DivisorSigma(21,n)/n^21): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Numerator[DivisorSigma[21, n]/n^21], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, numerator(sigma(n, 21)/n^21)) \\ G. C. Greubel, Nov 05 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017706(n) = zeta(21).
Dirichlet g.f. of a(n)/A017706(n): zeta(s)*zeta(s+21).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017706(k) = zeta(22). (End)

A017706 Denominator of sum of -21st powers of divisors of n.

Original entry on oeis.org

1, 2097152, 10460353203, 4398046511104, 476837158203125, 609359740010496, 558545864083284007, 9223372036854775808, 109418989131512359209, 500000000000000000000, 7400249944258160101211, 11501279977342425366528
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. A017705.

Programs

  • Magma
    [Denominator(DivisorSigma(21,n)/n^21): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Denominator[DivisorSigma[21, n]/n^21], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 21)/n^21)) \\ G. C. Greubel, Nov 05 2018

A017707 Numerator of sum of -22nd powers of divisors of n.

Original entry on oeis.org

1, 4194305, 31381059610, 17592190238721, 2384185791015626, 65810867613760525, 3909821048582988050, 73786993887028445185, 984770902214992292491, 1000000238418579520993, 81402749386839761113322, 92010261758627305193135, 3211838877954855105157370, 8199490986588434846527625
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. A017708 (denominator).

Programs

  • Magma
    [Numerator(DivisorSigma(22,n)/n^22): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Numerator[DivisorSigma[22, n]/n^22], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
DivisorSigma[-22,Range[20]]//Numerator (* Harvey P. Dale, Sep 19 2023 *)
  • PARI
    vector(20, n, numerator(sigma(n, 22)/n^22)) \\ G. C. Greubel, Nov 05 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017708(n) = zeta(22).
Dirichlet g.f. of a(n)/A017708(n): zeta(s)*zeta(s+22).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017708(k) = zeta(23). (End)

A017708 Denominator of sum of -22nd powers of divisors of n.

Original entry on oeis.org

1, 4194304, 31381059609, 17592186044416, 2384185791015625, 65810851921133568, 3909821048582988049, 73786976294838206464, 984770902183611232881, 1000000000000000000000, 81402749386839761113321
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. A017707.

Programs

  • Magma
    [Denominator(DivisorSigma(22,n)/n^22): n in [1..20]]; // G. C. Greubel, Nov 05 2018
  • Mathematica
    Table[Denominator[DivisorSigma[22, n]/n^22], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 22)/n^22)) \\ G. C. Greubel, Nov 05 2018

A017709 Numerator of sum of -23rd powers of divisors of n.

Original entry on oeis.org

1, 8388609, 94143178828, 70368752566273, 11920928955078126, 65810859767097521, 27368747340080916344, 590295880727458217985, 8862938119746644274757, 50000005960464481733367, 895430243255237372246532, 1656184514187480740117011, 41753905413413116367045798
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. A017710 (denominator).

Programs

  • Magma
    [Numerator(DivisorSigma(23,n)/n^23): n in [1..20]]; // G. C. Greubel, Nov 03 2018
  • Mathematica
    Table[Numerator[Total[Divisors[n]^-23]],{n,12}] (* Harvey P. Dale, Oct 19 2012 *)
Table[Numerator[DivisorSigma[23, n]/n^23], {n, 1, 20}] (* G. C. Greubel, Nov 03 2018 *)
  • PARI
    a(n) = numerator(sigma(n, 23)/n^23); \\ G. C. Greubel, Nov 03 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017710(n) = zeta(23).
Dirichlet g.f. of a(n)/A017710(n): zeta(s)*zeta(s+23).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017710(k) = zeta(24). (End)

A017710 Denominator of sum of -23rd powers of divisors of n.

Original entry on oeis.org

1, 8388608, 94143178827, 70368744177664, 11920928955078125, 65810851921133568, 27368747340080916343, 590295810358705651712, 8862938119652501095929, 50000000000000000000000, 895430243255237372246531
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. A017709.

Programs

  • Magma
    [Denominator(DivisorSigma(23,n)/n^23): n in [1..20]]; // G. C. Greubel, Nov 03 2018
  • Mathematica
    Table[Denominator[DivisorSigma[23, n]/n^23], {n, 1, 20}] (* G. C. Greubel, Nov 03 2018 *)
  • PARI
    a(n) = denominator(sigma(n, 23)/n^23); \\ G. C. Greubel, Nov 03 2018

A017711 Numerator of sum of -24th powers of divisors of n.

Original entry on oeis.org

1, 16777217, 282429536482, 281474993487873, 59604644775390626, 2369190810383965297, 191581231380566414402, 4722366764344638701569, 79766443077154939399843, 500000029802322396083921, 9849732675807611094711842, 13249475323675656646347131, 542800770374370512771595362
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. A017712 (denominator).

Programs

  • Magma
    [Numerator(DivisorSigma(24,n)/n^24): n in [1..20]]; // G. C. Greubel, Nov 03 2018
  • Mathematica
    Table[Numerator[DivisorSigma[24, n]/n^24], {n, 1, 20}] (* G. C. Greubel, Nov 03 2018 *)
  • PARI
    a(n) = numerator(sigma(n, 24)/n^24); \\ Michel Marcus, Nov 01 2013

Formula

Dirichlet g.f.: zeta(s)*zeta(s+24) (for fraction A017711/A017712). - Franklin T. Adams-Watters, Sep 11 2005
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017712(n) = zeta(24).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017712(k) = zeta(25). (End)
Previous Showing 81-90 of 97 results. Next

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

Content is available under The OEIS End-User License Agreement.