A230369 A strong divisibility sequence associated with the algebraic integer 2 + i.

1, 2, 1, 8, 1, 2, 1, 48, 1, 2, 1, 104, 1, 2, 1, 1632, 1, 2, 1, 8, 1, 2, 1, 1872, 1, 2, 109, 232, 1, 1342, 1, 3264, 1, 2, 1, 3848, 149, 2, 1, 1968, 1, 2, 1, 712, 1, 2, 1, 445536, 1, 2, 1, 424, 1, 218, 1, 1392, 1, 2, 1, 69784, 1, 2, 1, 6528, 1, 2, 1, 8, 1, 2, 1, 15168816, 1, 298, 1, 8, 1, 2, 1, 66912, 109, 2

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 + i. For other examples see A230368, A235450 and (conjecturally) A082630.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• 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, A235450.

Programs

Maple

seq( gcd( 1/2*((2 - I)^n + (2 + I)^n - 2), I/2*((2 + I)^n - (2 - I )^n) ), n = 1..80 );

Mathematica

Table[GCD[((2-I)^n +(2+I)^n -2)/2, I*((2+I)^n -(2-I)^n)/2], {n, 0, 85}] (* G. C. Greubel, Mar 21 2019 *)

PARI

{a(n) = gcd(((2-I)^n +(2+I)^n -2)/2, I*((2+I)^n -(2-I)^n)/2)}; \\ G. C. Greubel, Mar 21 2019

Formula

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

a(n) = gcd(((2 - i)^n + (2 + i)^n - 2)/2, i*((2 + i)^n - (2 - i)^n)/2).

As n -> inf, lim sup log(a(n))/n = 0.