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

2, 1, 1, 2, 2, 4, 6, 8, 6, 8, 14, 25, 36, 180, 76, 80, 66, 2068, 354, 7316, 740, 1776, 2198, 264, 792, 3278, 122396, 848, 17312, 27743, 36696, 17896832

Offset: 0

Author

M. F. Hasler, Apr 29 2007, corrected May 19 2007

Keywords

• more
• nonn

Comments

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.

Examples

			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).
		

Links

• Troy Vasiga and Jeffrey Shallit, On the iteration of certain quadratic maps over GF(p), Discrete Math. 277 (2004) 219-240 (see also doi).

Crossrefs

Cf. A003010.

Programs

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; }
A128976=vector(20,i,numcycles(i-1))

Formula

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) }

Extensions

a(20)-a(31) from Max Alekseyev, Aug 25 2023