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.

Showing 1-4 of 4 results.

A091258 Denominator of sigma_3(n)/sigma(n).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 31, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 1, 1, 1, 1, 1, 7, 13, 1, 1, 1, 19, 31, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 7, 1, 1, 1, 1, 1, 19, 13
Offset: 1

Views

Author

Labos Elemer, Feb 12 2004

Keywords

Comments

Conjecture: The set of distinct terms is A004611. - Michel Marcus, Aug 10 2024

Links

Crossrefs

Cf. A001158, A000203, A091259 (numerators), A091260.
Cf. A002476, A004611.

Programs

  • Mathematica
    Denominator[Table[DivisorSigma[3,n]/DivisorSigma[1,n],{n,100}]] (* Harvey P. Dale, Oct 06 2017 *)
  • PARI
    a(n) = denominator(sigma(n, 3)/sigma(n)); \\ Michel Marcus, Dec 29 2013

A091259 Numerator of sigma_3(n)/sigma(n).

Original entry on oeis.org

1, 3, 7, 73, 21, 21, 43, 39, 757, 63, 111, 73, 157, 129, 147, 151, 273, 2271, 343, 219, 301, 333, 507, 273, 15751, 471, 511, 3139, 813, 441, 931, 4161, 777, 819, 903, 55261, 1333, 1029, 1099, 819, 1641, 903, 1807, 8103, 15897, 1521, 2163, 1057, 39331, 47253
Offset: 1

Views

Author

Labos Elemer, Feb 12 2004

Keywords

Links

Crossrefs

Cf. A001158, A000203, A002061, A002117, A086463, A091258 (denominators), A091260.
Cf. A032766.

Programs

  • Magma
    [Numerator(DivisorSigma(3,n)/DivisorSigma(1,n)): n in [1..50]]; // Vincenzo Librandi, Jan 26 2018
  • Maple
    seq(numer(numtheory:-sigma[3](n)/numtheory:-sigma(n)),n=1..100); # Robert Israel, Jan 25 2018
  • Mathematica
    Array[Numerator[DivisorSigma[3,#]/DivisorSigma[1,#]]&,50] (* Harvey P. Dale, Feb 29 2016 *)
  • PARI
    a(n) = numerator(sigma(n, 3)/sigma(n)); \\ Michel Marcus, Jan 26 2018

Formula

a(p) = A002061(p), for prime p. - Robert Israel, Jan 25 2018
Sum_{k=1..n} a(k)/A091258(k) ~ c * n^3, where c = (Pi^2/18)*zeta(3)^2 * Product_{p prime} (1 - 2/p^2 - 1/p^3 + 5/p^5 - 3/p^6) = 0.2382648075... . - Amiram Eldar, Nov 21 2022
Conjecture: a(n) (mod 3) = A353816(n). - Michel Marcus, Aug 11 2024

A298754 Numerator of sigma_3(n)/sigma_2(n).

Original entry on oeis.org

1, 9, 14, 73, 63, 126, 172, 117, 757, 567, 666, 146, 1099, 1548, 882, 151, 2457, 6813, 3430, 219, 2408, 5994, 6084, 1638, 15751, 9891, 1022, 12556, 12195, 7938, 14896, 12483, 9324, 22113, 10836, 55261, 25327, 6174, 15386, 567, 34461, 21672, 39754, 16206, 6813, 54756, 51912, 2114, 39331, 47253
Offset: 1

Views

Author

Robert Israel, Jan 26 2018

Keywords

Links

Crossrefs

Cf. A001158, A001157, A000203, A091258, A091259, A091260 (denominator).

Programs

  • Maple
    seq(numer(numtheory:-sigma[3](n)/numtheory:-sigma[2](n)),n=1..100);
  • Mathematica
    a[n_] := Numerator[DivisorSigma[3, n]/DivisorSigma[2, n]]; Array[a, 100] (* Amiram Eldar, Dec 20 2024 *)
  • PARI
    a(n) = numerator(sigma(n, 3)/sigma(n, 2)); \\ Michel Marcus, Jan 27 2018

A379814 a(n) = sigma_2(n) * sigma_3(n).

Original entry on oeis.org

1, 45, 280, 1533, 3276, 12600, 17200, 49725, 68887, 147420, 162504, 429240, 373660, 774000, 917280, 1596221, 1425060, 3099915, 2483320, 5022108, 4816000, 7312680, 6449040, 13923000, 10253901, 16814700, 16760800, 26367600, 20536380, 41277600, 28659904, 51117885
Offset: 1

Views

Author

Amiram Eldar, Jan 03 2025

Keywords

Comments

See A379812 for more links and Ramanujan's general formula.

References

  • Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Links

Crossrefs

Cf. A001157, A001158, A091260, A092344, A092344, A298754, A379812, A379813.
Cf. A002117, A013662, A013664, A013665.

Programs

  • Mathematica
    a[n_] := Times @@ DivisorSigma[{2, 3}, n]; Array[a, 50]
  • PARI
    a(n) = {my(f = factor(n)); sigma(f, 2) * sigma(f, 3);}

Formula

a(n) = A001157(n) * A001158(n).
Multiplicative with a(p^e) = (p^(2*e+2)-1) * (p^(3*e+3)-1) / ((p^2-1) * (p^3-1)).
Dirichlet g.f.: zeta(s) * zeta(s-2) * zeta(s-3) * zeta(s-5) / zeta(2*s-5).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(3) * zeta(4) * zeta(6) / zeta(7) = Pi^10 * zeta(3) / (85050 * zeta(7)) = 1.31261826893951336264... .
Showing 1-4 of 4 results.

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