OFFSET
1,4
COMMENTS
a(n) is a simple (the simplest?) example of a divisibility sequence associated to a rational point on an algebraic group of dimension larger than two. Specifically, it is the divisibility sequence associated to the point (2,3) on the two-dimensional torus G_m^2. Ailon and Rudnick conjecture that a(n) = 1 for infinitely many n.
According to Corvaja, a(n) < 2^n - 1 for all but finitely many n.
REFERENCES
Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of a^n-1 and b^n-1. Math. Z. 243 (2003), no. 1, 79-84
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
N. Ailon, Z. Rudnick, Torsion points on curves and common divisors of a^k-1 and b^k-1, Acta Arith. 113 (2004), no. 1, 31-38.
Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of a^n-1 and b^n-1, Math. Z. 243 (2003), no. 1, 79-84
P. Corvaja, Greatest Common Divisors in Vojta's Conjecture: Arithmetic and Geometry, Journées Arithmétiques 2011.
FORMULA
a(n) = gcd(2^n - 1, 3^n - 1).
MAPLE
seq(igcd(2^n-1, 3^n-1), n=1..100); # Robert Israel, Sep 02 2015
MATHEMATICA
Table[GCD[2^n - 1, 3^n - 1], {n, 100}] (* Vincenzo Librandi, Sep 02 2015 *)
PROG
(PARI) vector(100, n, gcd(2^n-1, 3^n-1))
(Haskell)
a086892 n = a086892_list !! (n-1)
a086892_list = tail $ zipWith gcd a000225_list a003462_list
-- Reinhard Zumkeller, Jul 18 2015
(Magma) [Gcd(2^n-1, 3^n-1): n in [1..75]]; // Vincenzo Librandi, Sep 02 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Joseph H. Silverman (jhs(AT)math.brown.edu), Sep 18 2003
EXTENSIONS
Replaced arXiv URL with non-cached version by R. J. Mathar, Oct 23 2009
STATUS
approved