A235450 A strong divisibility sequence associated with the algebraic integer 2 + 3*sqrt(3).

1, 6, 13, 24, 1, 234, 1, 48, 13, 66, 1, 34632, 1, 6, 13, 96, 1, 702, 1, 264, 13, 6, 1, 346320, 1, 6, 13, 24, 59, 2574, 1, 192, 13, 6, 71, 7584408, 1, 6, 169, 16368, 1, 234, 1, 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

Offset: 1

Author

Peter Bala, Jan 10 2014

Keywords

• nonn
• easy

Comments

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.

Links

• J. H. Silverman, Divisibility sequences and powers of algebraic integers, Documenta Mathematica, Extra Volume: John H. Coates' Sixtieth Birthday (2006) 711-727

Crossrefs

Cf. A082630, A230368, A230369.

Programs

Maple

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);

Formula

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

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)) ).