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A128976 Number of cycles for the map LL:x->x^2-2 acting on Z/(2^n-1). 2

%I

%S 2,1,1,2,2,4,6,8,6,8,14,25,36,180,76,80,66,2068,354,7316

%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) such x=LL^c(x) for some positive integer c, i.e. { x, LL(x), ..., LL^c(x)=x }.

%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) 219-240.

%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) = { local(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 } 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

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Last modified May 31 19:11 EDT 2020. Contains 334748 sequences. (Running on oeis4.)