A235450 - OEIS

%I #18 Mar 21 2019 13:30:34

%S 1,6,13,24,1,234,1,48,13,66,1,34632,1,6,13,96,1,702,1,264,13,6,1,

%T 346320,1,6,13,24,59,2574,1,192,13,6,71,7584408,1,6,169,16368,1,234,1,

%U 24,13,282,1,4848480,1,66,13,24,1,2106,1,48,13,354,1,23238072,1,6,13,384,1,234,1,24,13,4686,1

%N A strong divisibility sequence associated with the algebraic integer 2 + 3*sqrt(3).

%C Let alpha be an algebraic integer and define a sequence of integers a(n) by the condition a(n) = max {integer d : alpha^n == 1 (mod d)}. Silverman shows that a(n) is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all n and m in N; in particular, if n divides m then a(n) divides a(m). For the present sequence we take alpha = 2 + 3*sqrt(3). For other examples see A230368, A230369 and (conjecturally) A082630.

%H J. H. Silverman, <a href="https://www.math.uni-bielefeld.de/documenta/vol-coates/silverman.pdf">Divisibility sequences and powers of algebraic integers</a>, Documenta Mathematica, Extra Volume: John H. Coates' Sixtieth Birthday (2006) 711-727

%F a(n) = max {integer d : (2 + 3*sqrt(3))^n  == 1 (mod d)}.

%F a(n) = gcd( 1/2*((2 - 3*sqrt(3))^n + (2 + 3*sqrt(3))^n - 2), ((2 + 3*sqrt(3))^n - (2 - 3*sqrt(3))^n)/(2*sqrt(3)) ).

%p seq(gcd( expand(1/2*((2 - 3*sqrt(3))^n + (2 + 3*sqrt(3))^n - 2)), expand(((2 + 3*sqrt(3))^n - (2 - 3*sqrt(3))^n)/(2*sqrt(3))) ), n = 1 .. 80);

%Y Cf. A082630, A230368, A230369.

%K nonn,easy

%O 1,2

%A _Peter Bala_, Jan 10 2014