0,1

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 }.

Troy Vasiga and Jeffrey Shallit: On the iteration of certain quadratic maps over GF(p), Discrete Math. (277) 219-240.

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) }

a(0)=2 since fixed points 2 and -1 are the only cycles for LL on Z/(0) = Z;

a(1)=1 since Z/(1) = {0};

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;

a(3)=2 since 3,4(=-3) -> 0 -> 5(=-2) -> {2} and 1 -> {6(=-1)} for LL acting on Z/(7);

a(5)=4 since {2}, {30}, {12,18} and {3,7,16,6} are the cycles for LL acting on Z/(31).

(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))

Cf. A003010.

Adjacent sequences: A128973 A128974 A128975 this_sequence A128977 A128978 A128979

Sequence in context: A045870 A036863 A083698 this_sequence A153902 A046772 A114551

more,nonn

M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 29 2007, corrected May 19 2007