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 1-3 of 3 results.

A078166 Numbers k such that phi(k) is a perfect sixth power.

Original entry on oeis.org

1, 2, 85, 128, 136, 160, 170, 192, 204, 240, 4369, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 15360, 15420, 16320, 47197, 47239, 47989, 49267, 49589, 50557, 51319, 52429, 52649, 55699, 57589, 57953, 59495, 63973
Offset: 1

Views

Author

Labos Elemer, Nov 27 2002

Keywords

Comments

As phi(2^(6*n+1)) = (2^n)^6, A277757 is a subsequence. - Bernard Schott, Sep 23 2022

Examples

			phi of the sequence includes 1, 64, 4096, 46656,..; powers arise several times; a(3)= A053576(6) = 85; in sequence relatively large jumps are observable when power of new numbers appear.

Links

Crossrefs

A277757 is a subsequence.
Numbers k such that phi(k) is a perfect power: A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th, this sequence), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power).
Cf. A001317, A053576, A045544, A000010.

Programs

A381195 Expansion of g.f. (1 - sqrt(1 - 1728*x))/(864*x).

Original entry on oeis.org

1, 432, 373248, 403107840, 487599243264, 631928619270144, 857978513934778368, 1204601833564428828672, 1734626640332777513287680, 2547819609320783611516944384, 3802273336964543978787469000704, 5749037285490390495926653129064448, 8788066841328079995004188536982208512
Offset: 0

Views

Author

Stefano Spezia, Feb 16 2025

Keywords

Links

Crossrefs

Cf. A009971, A277757.

Programs

  • Mathematica
    CoefficientList[Series[(1-Sqrt[1-1728x])/(864x),{x,0,12}],x]

Formula

a(n) = (-27)^n*2^(1+6*n)*binomial(1/2,1+n).
E.g.f.: exp(864*x)*(BesselI(0, 864*x) - BesselI(1, 864*x)).
D-finite with recurrence (n+1)*a(n) +864*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 18 2025
a(n) ~ 1728^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, May 29 2025

A381292 Expansion of g.f. hypergeom([1/2, 3/4, 5/4], [1, 3/2], 256*x).

Original entry on oeis.org

1, 80, 12096, 2196480, 435635200, 91017658368, 19681596211200, 4361120388218880, 984138122900275200, 225245492144504832000, 52138539404512009912320, 12180522019129546663526400, 2867511425916768698757021696, 679455041354637369514813030400, 161892954188496214335204360192000
Offset: 0

Views

Author

Stefano Spezia, Feb 19 2025

Keywords

Links

Crossrefs

Cf. A001044, A277757.

Programs

  • Mathematica
    a[n_]:=2^(6n+1)Gamma[2n+3/2]/((2n+1)Sqrt[Pi]n!^2); Array[a,15,0]

Formula

a(n) = 2^(6*n+1)*Gamma(2*n+3/2)/((2*n+1)*sqrt(Pi)*(n!)^2).
Showing 1-3 of 3 results.

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

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