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A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind. 0
2, 3, 17, 103, 163, 373, 487, 1733, 3469, 4373, 10259, 35153 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This is the last sequence on p. 15 of Smyth. The Lucas sequence with P = 3, Q = 5 is defined as v=2,3,-1,-18,-49,-57,.. where v(n) = P*v(n-1)-Q*v(n-2), with g.f. (2-3x)/(1-3x+5x^2).
The indices n such that n|v(n) define the sequence T = 1,3,9,27,81,153,243,459,... as listed by Smyth.
The OEIS sequence shows all distinct prime factors of elements of T.
LINKS
Richard André-Jeannin, Divisibility of generalized Fibonacci and Lucas numbers by their subscripts, Fibonacci Quart., 29(4) (1991) 364-366.
Yu. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539 (2001) 75-122.
R. D. Carmichael, On the numerical factors of the arithmetic forms alpha*n+-beta*n, Annals of Math., 2nd ser., 15 (1/4) (1913/14) 30-48.
Chris Smyth, The terms in Lucas sequences divisible by their indices, arXiv:0908.3832 [math.NT], Aug 28 2009.
MAPLE
a := {2} ;
v := proc(n)
option remember;
if n <= 1 then
op(n+1, [2, 3]) ;
else
3*procname(n-1)-5*procname(n-2) ;
end if;
end proc:
for n from 1 do
if modp(v(n), n) = 0 then
a := a union numtheory[factorset](n) ;
print(a);
end if;
end do: # R. J. Mathar, Jul 09 2013
MATHEMATICA
nmax = 10^7; v1 = 2; v2 = 3; s = {2}; For[n = 2, n <= nmax, n++, v3 = 3*v2 - 5*v1; v1 = v2; v2 = v3; If[Divisible[v3, n], u = Union[s, FactorInteger[n][[All, 1]] ]; If[u != s, s = u; Print["n = ", n, ", s = ", s]]]]; s (* Jean-François Alcover, Dec 08 2017 *)
CROSSREFS
Sequence in context: A245799 A056794 A135726 * A259535 A328340 A042978
KEYWORD
more,nonn
AUTHOR
Jonathan Vos Post, Aug 26 2009
EXTENSIONS
More detailed definition, comments rephrased, non-ascii characters in URL's removed - R. J. Mathar, Sep 09 2009
a(9)-a(12) from Jean-François Alcover, Dec 08 2017
STATUS
approved

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Last modified June 15 05:55 EDT 2024. Contains 373402 sequences. (Running on oeis4.)