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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A175180 Numbers k such that k^2 + 2 is powerful in the sense of A001694.

Original entry on oeis.org

5, 265, 13775, 716035, 9980583, 37220045, 1934726305
Offset: 1

Views

Author

Michel Lagneau, Mar 01 2010

Keywords

Comments

This sequence is infinite (F. Luca in De Koninck).
The values of k^2 are a subset of A076445, so 23 terms of the sequence are known from there. - R. J. Mathar, Mar 05 2010
Together with 1, supersequence of A238799. - Arkadiusz Wesolowski, Mar 06 2014
From Amiram Eldar, Feb 23 2024: (Start)
a(8) <= 100568547815.
A041042(2*k) is a term for all k >= 0 (since 3^3 * A041043(n)^2 - A041042(n)^2 = -1 if n is odd and 2 if n is even). (End)

Examples

			5 is in the sequence because 5^2 + 2 = 3^3 is powerful.
265 is in the sequence because 265^2 + 2 = 51^2*3^3 is powerful.
13775 is in the sequence because 13775^2 + 2 = 2651^2 * 3^3 is powerful.

References

  • Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 265, p. 71, Ellipses, Paris, 2008.

Links

Crossrefs

Cf. A001694, A041042, A041043, A076445, A238799.

Programs

  • Mathematica
    q[n_] := AllTrue[FactorInteger[n^2+2][[;;, 2]], # > 1 &]; Select[Range[10^6], q] (* Amiram Eldar, Feb 23 2024 *)
  • PARI
    is(n)=ispowerful(n^2+2) \\ Charles R Greathouse IV, Feb 04 2013

Extensions

Examples rephrased by R. J. Mathar, Feb 24 2010, Mar 05 2010
a(7) from Amiram Eldar, Feb 23 2024

A378683 a(0) = 1, a(n+1) = 6*a(n)^3 - 3*a(n).

Original entry on oeis.org

1, 3, 153, 21489003, 59538796254981950751153, 1266343134315970349117919634635229303292221774134557782012151266098003
Offset: 0

Views

Author

Robert FERREOL, Dec 03 2024

Keywords

Comments

If we define u(0) = 1 , u(n+1) = (u(n)/3)*(u(n)^2+9) / (u(n)^2 + 1), then u(n) = A238799(n) / a(n) ; this is Halley's method to calculate sqrt(3).

Links

Crossrefs

Cf. A002530, A002605, A238799.

Programs

  • Maple
    a:=1 : A:=NULL : for k to 5 do a:=6*a^3-3*a : A:=A,a od : A;
  • Mathematica
    NestList[6#^3-3# &, 1, 5] (* Stefano Spezia, Dec 06 2024 *)

Formula

a(n) = ((1 + sqrt(3))^(3^n) - (1 - sqrt(3))^(3^n))/2^((3^n+1)/2) / sqrt(3).
a(n) = A002530(3^n).
a(n) = A002605(3^n)/2^((3^n+1)/2).
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