cp's OEIS Frontend

A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind.

%I A164816 #32 Jan 05 2025 19:51:39
%S A164816 3,17,103,163,373,487,1733,3469,4373,8803,10259,15607,16069,26237,
%T A164816 26297,31193,31517,35153,37987,38047,38149,39367,52817,60427,60589,
%U A164816 61553,74357,76837,78713,100733,103979,114377,119891,152189,181277,231131,235891,238307,239783,280927,289243,316903,338581
%N A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind.
%C A164816 This is the last sequence on p. 15 of Smyth. [WARNING: Smyth lists 2 as a possible prime factor, which, in fact, is not possible. - _Max Alekseyev_, Sep 17 2024]
%C A164816 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).
%C A164816 The indices n such that n|v(n) define the sequence T = 1,3,9,27,81,153,243,459,... as listed by Smyth.
%C A164816 The OEIS sequence shows all distinct prime factors of elements of T.
%H A164816 Max Alekseyev, <a href="/A164816/b164816.txt">Table of n, a(n) for n = 1..627</a>
%H A164816 Richard André-Jeannin, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/29-4/andre-jeannin2.pdf">Divisibility of generalized Fibonacci and Lucas numbers by their subscripts</a>, Fibonacci Quart., 29(4) (1991) 364-366.
%H A164816 Yu. Bilu, G. Hanrot, and P. M. Voutier, <a href="http://dx.doi.org/10.1515/crll.2001.080">Existence of primitive divisors of Lucas and Lehmer numbers</a>, J. Reine Angew. Math., 539 (2001) 75-122.
%H A164816 R. D. Carmichael, <a href="http://www.jstor.org/stable/1967797">On the numerical factors of the arithmetic forms alpha*n+-beta*n</a>, Annals of Math., 2nd ser., 15 (1/4) (1913/14) 30-48.
%H A164816 Chris Smyth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Smyth/smyth2.html">The Terms in Lucas Sequences Divisible by their Indices</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4. Preprint: <a href="http://arxiv.org/abs/0908.3832">arXiv:0908.3832 [math.NT]</a>, 2009.
%Y A164816 Cf. A057719, A066364, A087807.
%K A164816 nonn
%O A164816 1,1
%A A164816 _Jonathan Vos Post_, Aug 26 2009
%E A164816 More detailed definition, comments rephrased, non-ascii characters in URL's removed - _R. J. Mathar_, Sep 09 2009
%E A164816 a(8)-a(9), a(11), a(18) from _Jean-François Alcover_, Dec 08 2017
%E A164816 Incorrect codes (depending on a search limit) removed, prime 2 removed, terms a(10), (12)-a(17), and a(19) onward added by _Max Alekseyev_, Sep 17 2024