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 21-26 of 26 results.

A126824 Ramanujan numbers (A000594) read mod 16384.

Original entry on oeis.org

1, 16360, 252, 14912, 4830, 10336, 16024, 2560, 1045, 15152, 10324, 5888, 12086, 8640, 4744, 4096, 8114, 7688, 11820, 896, 7584, 14368, 14664, 6144, 10663, 4848, 6552, 5632, 5222, 832, 11616, 0, 12976, 1872, 14288, 1856, 9534, 11232, 14632, 11264, 2938, 14592
Offset: 1

Views

Author

N. J. A. Sloane, Feb 25 2007

Keywords

References

  • Oddmund Kolberg, Congruences for Ramanujan's Function ̈tau(n), Univ. Bergen Årbok Naturvit Rekke, No. 11, 1962.

Links

Crossrefs

Cf. A000594, A013959.

Programs

  • Mathematica
    a[n_] := Mod[RamanujanTau[n], 16384]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)
  • PARI
    a(n) = ramanujantau(n) % 16384; \\ Amiram Eldar, Jan 05 2025

Formula

a(n) == 705 * sigma_11(n) (mod 16384) for n == 7 (mod 8) (Kolberg, 1962). - Amiram Eldar, Jan 05 2025

A126829 Ramanujan numbers (A000594) read mod 243.

Original entry on oeis.org

1, 219, 9, 229, 213, 27, 23, 159, 81, 234, 12, 117, 116, 177, 216, 70, 126, 0, 38, 177, 207, 198, 69, 216, 223, 132, 0, 164, 84, 162, 98, 9, 108, 135, 39, 81, 236, 60, 72, 90, 87, 135, 203, 75, 0, 45, 75, 144, 21, 237, 162, 77, 135, 0, 126, 12, 99, 171, 24, 135, 50, 78, 162
Offset: 1

Views

Author

N. J. A. Sloane, Feb 25 2007

Keywords

References

  • Oddmund Kolberg, Congruences for Ramanujan's Function ̈tau(n), Univ. Bergen Årbok Naturvit Rekke, No. 11, 1962.

Links

Crossrefs

Cf. A000594, A013959.

Programs

  • Mathematica
    a[n_] := Mod[RamanujanTau[n], 243]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)
  • PARI
    a(n) = ramanujantau(n) % 243; \\ Amiram Eldar, Jan 05 2025

Formula

a(n) == sigma_11(n) (mod 243) for n == 1 (mod 3) (Kolberg, 1962). - Amiram Eldar, Jan 05 2025

A126830 Ramanujan numbers (A000594) read mod 729.

Original entry on oeis.org

1, 705, 252, 715, 456, 513, 23, 645, 81, 720, 255, 117, 359, 177, 459, 70, 612, 243, 524, 177, 693, 441, 555, 702, 466, 132, 0, 407, 570, 648, 584, 495, 108, 621, 282, 324, 236, 546, 72, 333, 573, 135, 689, 75, 486, 531, 75, 144, 264, 480, 405, 77, 135, 0, 369, 255
Offset: 1

Views

Author

N. J. A. Sloane, Feb 25 2007

Keywords

References

  • M. H. Ashworth, Congruence and identical properties of modular forms, Diss. University of Oxford, 1968.
  • Oddmund Kolberg, Congruences for Ramanujan's Function ̈tau(n), Univ. Bergen Årbok Naturvit Rekke, No. 11, 1962.

Links

Crossrefs

Cf. A000594, A013959.

Programs

  • Mathematica
    a[n_] := Mod[RamanujanTau[n], 729]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)
  • PARI
    a(n) = ramanujantau(n) % 729; \\ Amiram Eldar, Jan 05 2025

Formula

From Amiram Eldar, Jan 05 2025: (Start)
a(n) == 53 * sigma_11(n) (mod 729) for n == 2 (mod 3) (Kolberg, 1962).
a(n) == n^(-620) * sigma_1231(n) for n == 1 (mod 3) (Ashworth, 1968). (End)

A169976 Expansion of (psi(x)^24 + psi(-x)^24) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 276, 11178, 177400, 1612875, 10131156, 48897678, 193740408, 658523925, 1980143600, 5386270686, 13477895856, 31425764410, 68969957700, 143635113000, 285718115112, 545796171084, 1005775268868, 1794713445350, 3111031518000
Offset: 0

Views

Author

Michael Somos, Aug 15 2010

Keywords

Examples

			1 + 276*x + 11178*x^2 + 177400*x^3 + 1612875*x^4 + 10131156*x^5 + ...
q^3 + 276*q^5 + 11178*q^7 + 177400*q^9 + 1612875*q^11 + 10131156*q^13 + ...

Links

Crossrefs

Cf. A000594, A013959, A014809.

Programs

  • Mathematica
    QP:= Pochhammer; a[n_]:= SeriesCoefficient[(QP[q, q])^24*(QP[-q^(1/2), q^(1/2)]^24 + QP[q^(1/2), -q^(1/2)]^24)/2, {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Apr 04 2018 *)
a[n_] := (DivisorSigma[11, 2*n+3] - RamanujanTau[2*n+3]) / 176896; Array[a, 20, 0] (* Amiram Eldar, Jan 11 2025 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^24, n))}
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 3; (sigma(n, 11) - polcoeff( x * eta(x + x * O(x^n))^24, n)) / 176896 )}

Formula

a(n) = (A013959(2*n + 3) - A000594(2*n + 3)) / 176896 = A014809(2*n).

A081867 a(n) = sigma_11(2n-1).

Original entry on oeis.org

1, 177148, 48828126, 1977326744, 31381236757, 285311670612, 1792160394038, 8649804864648, 34271896307634, 116490258898220, 350279478046112, 952809757913928, 2384185839843751, 5559091947792280, 12200509765705830, 25408476896404832, 50542391825574576, 96549159399201744
Offset: 1

Views

Author

Benoit Cloitre, Apr 12 2003

Keywords

Crossrefs

Cf. A013959.

Programs

  • Mathematica
    Table[DivisorSigma[11,2n-1],{n,20}] (* Harvey P. Dale, Jun 26 2012 *)
  • PARI
    a(n) = sigma(2*n-1, 11); \\ Amiram Eldar, Jan 08 2025

Formula

Sum_{k=1..n} a(k) ~ c * n^12, where c = 691 * Pi^12 / 3742200 = 170.666988... . - Amiram Eldar, Jan 07 2025

A094469 Numbers k such that sum of 11th powers of divisors of k is divisible by the square of Euler-phi of k.

Original entry on oeis.org

1, 2, 3, 6, 1645, 3290, 4935, 9870, 3831674, 11495022, 346014339, 692028678
Offset: 1

Views

Author

Labos Elemer, May 19 2004

Keywords

Crossrefs

Cf. A000010, A013959, A015763, A091285.

Programs

  • Mathematica
    Do[ If[ Mod[ DivisorSigma[11, n], EulerPhi[n]^2] == 0, Print[n]], {n, 10^7}] (* Robert G. Wilson v, May 23 2004 *)
  • PARI
    isok(k) = (sigma(k, 11) % eulerphi(k)^2) == 0; \\ Michel Marcus, Mar 07 2020

Formula

A013959(k)/A000010(k)^2 is an integer.

Extensions

a(9) from Robert G. Wilson v, May 23 2004
a(10) from Labos Elemer, May 26 2004
a(6) corrected and a(11)-a(12) added by Amiram Eldar, Mar 07 2020
Previous Showing 21-26 of 26 results.

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

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