Search: id:A128976
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%I A128976
%S A128976 2,1,1,2,2,4,6,8,6,8,14,25,36,180,76,80,66,2068,354,7316
%N A128976 Number of cycles for the map LL:x->x^2-2 acting on Z/(2^n-1).
%C A128976 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 A128976 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 A128976 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 A128976 a(0)=2 since fixed points 2 and -1 are the only cycles for LL on Z/(0) 
               = Z;
%e A128976 a(1)=1 since Z/(1) = {0};
%e A128976 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 A128976 a(3)=2 since 3,4(=-3) -> 0 -> 5(=-2) -> {2} and 1 -> {6(=-1)} for LL 
               acting on Z/(7);
%e A128976 a(5)=4 since {2}, {30}, {12,18} and {3,7,16,6} are the cycles for LL 
               acting on Z/(31).
%o A128976 (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 A128976 Cf. A003010.
%Y A128976 Sequence in context: A045870 A036863 A083698 this_sequence A153902 A046772 
               A114551
%Y A128976 Adjacent sequences: A128973 A128974 A128975 this_sequence A128977 A128978 
               A128979
%K A128976 more,nonn
%O A128976 0,1
%A A128976 M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 29 2007, corrected 
               May 19 2007

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