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

2, 3, 17, 103, 163, 373, 487, 1733, 3469, 4373, 10259, 35153

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

Table of n, a(n) for n=1..12.

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

Adjacent sequences: A164813 A164814 A164815 * A164817 A164818 A164819

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