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

3, 17, 103, 163, 373, 487, 1733, 3469, 4373, 8803, 10259, 15607, 16069, 26237, 26297, 31193, 31517, 35153, 37987, 38047, 38149, 39367, 52817, 60427, 60589, 61553, 74357, 76837, 78713, 100733, 103979, 114377, 119891, 152189, 181277, 231131, 235891, 238307, 239783, 280927, 289243, 316903, 338581

Offset: 1

Author

Jonathan Vos Post, Aug 26 2009

Keywords

• nonn

Comments

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]

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

• Max Alekseyev, Table of n, a(n) for n = 1..627
• 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, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4. Preprint: arXiv:0908.3832 [math.NT], 2009.

Crossrefs

Cf. A057719, A066364, A087807.

Extensions

More detailed definition, comments rephrased, non-ascii characters in URL's removed - R. J. Mathar, Sep 09 2009

a(8)-a(9), a(11), a(18) from Jean-François Alcover, Dec 08 2017

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