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.

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A017671 Numerator of sum of -4th powers of divisors of n.

Original entry on oeis.org

1, 17, 82, 273, 626, 697, 2402, 4369, 6643, 5321, 14642, 3731, 28562, 20417, 51332, 69905, 83522, 112931, 130322, 85449, 196964, 124457, 279842, 179129, 391251, 242777, 538084, 46839, 707282, 218161, 923522, 1118481, 1200644, 41761, 1503652, 604513, 1874162
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

Examples

			1, 17/16, 82/81, 273/256, 626/625, 697/648, 2402/2401, 4369/4096, 6643/6561, 5321/5000, ...

Links

Crossrefs

Cf. A017672 (denominator), A013662, A013663.

Programs

  • Magma
    [Numerator(DivisorSigma(4,n)/n^4): n in [1..40]]; // G. C. Greubel, Nov 08 2018
  • Mathematica
    Table[Numerator[DivisorSigma[-4, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[4, n]/n^4], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)
  • PARI
    vector(40, n, numerator(sigma(n, 4)/n^4)) \\ G. C. Greubel, Nov 08 2018

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^4*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017672(n) = zeta(4) (A013662).
Dirichlet g.f. of a(n)/A017672(n): zeta(s)*zeta(s+4).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017672(k) = zeta(5) (A013663). (End)

A017672 Denominator of sum of -4th powers of divisors of n.

Original entry on oeis.org

1, 16, 81, 256, 625, 648, 2401, 4096, 6561, 5000, 14641, 3456, 28561, 19208, 50625, 65536, 83521, 104976, 130321, 80000, 194481, 117128, 279841, 165888, 390625, 228488, 531441, 43904, 707281, 202500, 923521, 1048576, 1185921, 39304, 1500625, 559872, 1874161
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

Examples

			1, 17/16, 82/81, 273/256, 626/625, 697/648, 2402/2401, 4369/4096, 6643/6561, 5321/5000, ...

Links

Crossrefs

Cf. A017671.

Programs

  • Magma
    [Denominator(DivisorSigma(4,n)/n^4): n in [1..40]]; // G. C. Greubel, Nov 08 2018
  • Mathematica
    Table[Denominator[DivisorSigma[-4, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[4, n]/n^4], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)
  • PARI
    vector(40, n, denominator(sigma(n, 4)/n^4)) \\ G. C. Greubel, Nov 08 2018

Formula

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^4*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

A017673 Numerator of sum of -5th powers of divisors of n.

Original entry on oeis.org

1, 33, 244, 1057, 3126, 671, 16808, 33825, 59293, 51579, 161052, 64477, 371294, 69333, 254248, 1082401, 1419858, 652223, 2476100, 1652091, 4101152, 120789, 6436344, 687775, 9768751, 6126351, 14408200, 317251, 20511150, 349591, 28629152, 34636833, 13098896
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

Examples

			1, 33/32, 244/243, 1057/1024, 3126/3125, 671/648, 16808/16807, 33825/32768, 59293/59049, ...

Links

Crossrefs

Cf. A017674 (denominator), A013663, A013664.

Programs

  • Magma
    [Numerator(DivisorSigma(5,n)/n^5): n in [1..40]]; // G. C. Greubel, Nov 08 2018
  • Mathematica
    Table[Numerator[DivisorSigma[-5, n]], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
Table[Numerator[DivisorSigma[5, n]/n^5], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)
  • PARI
    vector(40, n, numerator(sigma(n, 5)/n^5)) \\ G. C. Greubel, Nov 08 2018

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^5*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017674(n) = zeta(5) (A013663).
Dirichlet g.f. of a(n)/A017674(n): zeta(s)*zeta(s+5).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017674(k) = zeta(6) (A013664). (End)

A017674 Denominator of sum of -5th powers of divisors of n.

Original entry on oeis.org

1, 32, 243, 1024, 3125, 648, 16807, 32768, 59049, 50000, 161051, 62208, 371293, 67228, 253125, 1048576, 1419857, 629856, 2476099, 1600000, 4084101, 117128, 6436343, 663552, 9765625, 5940688, 14348907, 307328, 20511149, 337500, 28629151, 33554432, 13045131
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

Examples

			1, 33/32, 244/243, 1057/1024, 3126/3125, 671/648, 16808/16807, 33825/32768, 59293/59049, ...

Links

Crossrefs

Cf. A017673.

Programs

  • Magma
    [Denominator(DivisorSigma(5,n)/n^5): n in [1..40]]; // G. C. Greubel, Nov 08 2018
  • Mathematica
    Table[Denominator[DivisorSigma[-5, n]], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
Table[Denominator[DivisorSigma[5, n]/n^5], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)
  • PARI
    vector(40, n, denominator(sigma(n, 5)/n^5)) \\ G. C. Greubel, Nov 08 2018

Formula

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^5*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

A017675 Numerator of sum of -6th powers of divisors of n.

Original entry on oeis.org

1, 65, 730, 4161, 15626, 23725, 117650, 266305, 532171, 101569, 1771562, 506255, 4826810, 3823625, 2281396, 17043521, 24137570, 34591115, 47045882, 32509893, 85884500, 57575765, 148035890, 97201325, 244156251, 12067025, 387952660, 244770825, 594823322
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

Examples

			1, 65/64, 730/729, 4161/4096, 15626/15625, 23725/23328, 117650/117649, 266305/262144, ...

Links

Crossrefs

Cf. A017676 (denominator), A013664, A013665.

Programs

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^6*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017676(n) = zeta(6) (A013664).
Dirichlet g.f. of a(n)/A017676(n): zeta(s)*zeta(s+6).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017676(k) = zeta(7) (A013665). (End)

A017676 Denominator of sum of -6th powers of divisors of n.

Original entry on oeis.org

1, 64, 729, 4096, 15625, 23328, 117649, 262144, 531441, 100000, 1771561, 497664, 4826809, 3764768, 2278125, 16777216, 24137569, 34012224, 47045881, 32000000, 85766121, 56689952, 148035889, 95551488, 244140625, 11881376, 387420489, 240945152, 594823321
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

Examples

			1, 65/64, 730/729, 4161/4096, 15626/15625, 23725/23328, 117650/117649, 266305/262144, ...

Links

Crossrefs

Cf. A017675.

Programs

Formula

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^6*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

A017677 Numerator of sum of -7th powers of divisors of n.

Original entry on oeis.org

1, 129, 2188, 16513, 78126, 23521, 823544, 2113665, 4785157, 5039127, 19487172, 9032611, 62748518, 13279647, 56979896, 270549121, 410338674, 205761751, 893871740, 645047319, 1801914272, 628461297, 3404825448, 385391585, 6103593751, 4047279411, 10465138360, 34691791
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

Examples

			1, 129/128, 2188/2187, 16513/16384, 78126/78125, 23521/23328, 823544/823543, 2113665/2097152, ...

Links

Crossrefs

Cf. A017678 (denominator), A013665, A013666.

Programs

  • Magma
    [Numerator(DivisorSigma(7,n)/n^7): n in [1..20]]; // G. C. Greubel, Nov 07 2018
  • Mathematica
    Table[Numerator[Total[Divisors[n]^-7]],{n,30}] (* Harvey P. Dale, Nov 29 2014 *)
Table[Numerator[DivisorSigma[7, n]/n^7], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    vector(20, n, numerator(sigma(n, 7)/n^7)) \\ G. C. Greubel, Nov 07 2018

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^7*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017678(n) = zeta(7) (A013665).
Dirichlet g.f. of a(n)/A017678(n): zeta(s)*zeta(s+7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017678(k) = zeta(8) (A013666). (End)

A017678 Denominator of sum of -7th powers of divisors of n.

Original entry on oeis.org

1, 128, 2187, 16384, 78125, 23328, 823543, 2097152, 4782969, 5000000, 19487171, 8957952, 62748517, 13176688, 56953125, 268435456, 410338673, 204073344, 893871739, 640000000, 1801088541, 623589472
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

Examples

			1, 129/128, 2188/2187, 16513/16384, 78126/78125, 23521/23328, 823544/823543, 2113665/2097152, ...

Links

Crossrefs

Cf. A017677.

Programs

  • Magma
    [Denominator(DivisorSigma(7,n)/n^7): n in [1..20]]; // G. C. Greubel, Nov 07 2018
  • Mathematica
    Table[Denominator[Total[Divisors[n]^-7]],{n,30}] (* Harvey P. Dale, Mar 21 2012 *)
Table[Denominator[DivisorSigma[7, n]/n^7], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 7)/n^7)) \\ G. C. Greubel, Nov 07 2018

Formula

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^7*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

A017679 Numerator of sum of -8th powers of divisors of n.

Original entry on oeis.org

1, 257, 6562, 65793, 390626, 843217, 5764802, 16843009, 43053283, 50195441, 214358882, 71955611, 815730722, 740777057, 2563287812, 4311810305, 6975757442, 11064693731, 16983563042, 12850228209, 37828630724, 27545116337, 78310985282, 55261912529, 152588281251
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

Examples

			1, 257/256, 6562/6561, 65793/65536, 390626/390625, 843217/839808, 5764802/5764801, 16843009/16777216, ...

Links

Crossrefs

Cf. A017680 (denominator), A013666, A013667.

Programs

  • Magma
    [Numerator(DivisorSigma(8,n)/n^8): n in [1..20]]; // G. C. Greubel, Nov 07 2018
  • Mathematica
    Table[Numerator[DivisorSigma[8, n]/n^8], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    vector(20, n, numerator(sigma(n, 8)/n^8)) \\ G. C. Greubel, Nov 07 2018

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^8*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017680(n) = zeta(8) (A013666).
Dirichlet g.f. of a(n)/A017680(n): zeta(s)*zeta(s+8).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017680(k) = zeta(9) (A013667). (End)

A017680 Denominator of sum of -8th powers of divisors of n.

Original entry on oeis.org

1, 256, 6561, 65536, 390625, 839808, 5764801, 16777216, 43046721, 50000000, 214358881, 71663616, 815730721, 737894528, 2562890625, 4294967296, 6975757441, 11019960576, 16983563041, 12800000000
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

Examples

			1, 257/256, 6562/6561, 65793/65536, 390626/390625, 843217/839808, 5764802/5764801, 16843009/16777216, ...

Links

Crossrefs

Cf. A017679.

Programs

  • Magma
    [Denominator(DivisorSigma(8,n)/n^8): n in [1..20]]; // G. C. Greubel, Nov 07 2018
  • Mathematica
    Table[Denominator[Total[1/Divisors[n]^8]],{n,20}] (* Harvey P. Dale, Dec 16 2013 *)
Table[Denominator[DivisorSigma[8, n]/n^8], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    vector(20, n, denominator(sigma(n, 8)/n^8)) \\ G. C. Greubel, Nov 07 2018

Formula

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^8*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
Previous Showing 31-40 of 72 results. Next

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

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