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%I #26 Aug 26 2023 04:35:55
%S 2,1,1,2,2,4,6,8,6,8,14,25,36,180,76,80,66,2068,354,7316,740,1776,
%T 2198,264,792,3278,122396,848,17312,27743,36696,17896832
%N Number of cycles for the map LL:x->x^2-2 acting on Z/(2^n-1).
%C A cycle is the orbit of an element x of Z/(2^n-1), i.e., { x, LL(x), ..., LL^c(x)=x } for some positive integer c.
%H Troy Vasiga and Jeffrey Shallit, <a href="http://www.cs.uwaterloo.ca/~tmjvasig/papers/newvasiga.pdf">On the iteration of certain quadratic maps over GF(p)</a>, Discrete Math. 277 (2004) 219-240 (see also <a href="https://doi.org/10.1016/S0012-365X(03)00158-4">doi</a>).
%F If p=2^n-1 is prime, then a(n) = 1/2 + Sum_{d|2^(n-1)-1} eulerphi(d)/ordp(2,d)/2, where ordp(2,d) = min { e in N* | 2^e == 1 (mod d) or 2^e == -1 (mod d) }
%e a(0)=2 since fixed points 2 and -1 are the only cycles for LL on Z/(0) = Z;
%e a(1)=1 since Z/(1) = {0};
%e a(2)=1 since 2=-1 is a cycle of length 1 (fixed point) for LL on Z/(3) and LL(0)=-2=1, LL(1)=-1;
%e a(3)=2 since 3,4(=-3) -> 0 -> 5(=-2) -> {2} and 1 -> {6(=-1)} for LL acting on Z/(7);
%e a(5)=4 since {2}, {30}, {12,18} and {3,7,16,6} are the cycles for LL acting on Z/(31).
%o (PARI) numcycles(q) = { my(Mq=2^q-1, v=vector(Mq+1), c=1, i, start, cyc=0); if(q<2,return(1+!q)); for( j=1, #v, if(v[j],next); i=j; start=c; until(v[i=1+((i-1)^2-2)%Mq],v[i]=c++); if(v[i]>start, cyc++)); cyc; }
%o A128976=vector(20,i,numcycles(i-1))
%Y Cf. A003010.
%K more,nonn
%O 0,1
%A _M. F. Hasler_, Apr 29 2007, corrected May 19 2007
%E a(20)-a(31) from _Max Alekseyev_, Aug 25 2023
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