A041042 -id:A041042 - cp's OEIS frontend

A041043 Denominators of continued fraction convergents to sqrt(27).

1, 5, 51, 260, 2651, 13515, 137801, 702520, 7163001, 36517525, 372338251, 1898208780, 19354426051, 98670339035, 1006057816401, 5128959421040, 52295652026801, 266607219555045, 2718367847577251, 13858446457441300

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,52,0,-1).

Cf. A010482, A041042.

Denominator[Convergents[Sqrt[27],50]] (* Harvey P. Dale, Apr 22 2012 *)
CoefficientList[Series[- (x^2 - 5 x - 1)/(x^4 - 52 x^2 + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 22 2013 *)
a0[n_] := (9+5*Sqrt[3]+(9-5*Sqrt[3])*(26+15*Sqrt[3])^(2*n))/(18*(26+15*Sqrt[3])^n) // Simplify
a1[n_] := (-1+(26+15*Sqrt[3])^(2*n))/(6*Sqrt[3]*(26+15*Sqrt[3])^n) // FullSimplify
Flatten[MapIndexed[{a0[#],a1[#]}&,Range[10]]] (* Gerry Martens, Jul 10 2015 *)

a(n) = 52*a(n-2)-a(n-4). G.f.: -(x^2-5*x-1)/(x^4-52*x^2+1). - Colin Barker, Jul 15 2012
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)]:
a0(n) = ((9+5*sqrt(3))/(26+15*sqrt(3))^n+(9-5*sqrt(3))*(26+15*sqrt(3))^n)/18.
a1(n) = (-1/(26+15*sqrt(3))^n+(26+15*sqrt(3))^n)/(6*sqrt(3)). (End)

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

Michel Lagneau, Mar 01 2010

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)

			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.

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

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 66.

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

q[n_] := AllTrue[FactorInteger[n^2+2][[;;, 2]], # > 1 &]; Select[Range[10^6], q] (* Amiram Eldar, Feb 23 2024 *)

is(n)=ispowerful(n^2+2) \\ Charles R Greathouse IV, Feb 04 2013

Examples rephrased by R. J. Mathar, Feb 24 2010, Mar 05 2010

a(7) from Amiram Eldar, Feb 23 2024

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A041043 Denominators of continued fraction convergents to sqrt(27).

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

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