cp's OEIS Frontend

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

%I A230369 #19 Mar 22 2019 00:29:51
%S A230369 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,
%T A230369 1,1342,1,3264,1,2,1,3848,149,2,1,1968,1,2,1,712,1,2,1,445536,1,2,1,
%U A230369 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
%N A230369 A strong divisibility sequence associated with the algebraic integer 2 + i.
%C A230369 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.
%H A230369 G. C. Greubel, <a href="/A230369/b230369.txt">Table of n, a(n) for n = 1..5000</a>
%H A230369 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 A230369 a(n) = max {integer d : (2 + i)^n == 1 (mod d)}.
%F A230369 a(n) = gcd(((2 - i)^n + (2 + i)^n - 2)/2, i*((2 + i)^n - (2 - i)^n)/2).
%F A230369 As n -> inf, lim sup log(a(n))/n = 0.
%p A230369 seq( gcd( 1/2*((2 - I)^n + (2 + I)^n - 2), I/2*((2 + I)^n - (2 - I )^n) ), n = 1..80 );
%t A230369 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 *)
%o A230369 (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
%Y A230369 Cf. A082630, A230368, A235450.
%K A230369 nonn,easy
%O A230369 1,2
%A A230369 _Peter Bala_, Jan 10 2014