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A128976 Number of cycles for the map LL:x->x^2-2 acting on Z/(2^n-1). 2
2, 1, 1, 2, 2, 4, 6, 8, 6, 8, 14, 25, 36, 180, 76, 80, 66, 2068, 354, 7316 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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 }.

LINKS

Table of n, a(n) for n=0..19.

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

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) }

EXAMPLE

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).

PROG

(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))

CROSSREFS

Cf. A003010.

Sequence in context: A036863 A209270 A083698 * A199627 A153902 A318205

Adjacent sequences:  A128973 A128974 A128975 * A128977 A128978 A128979

KEYWORD

more,nonn

AUTHOR

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

STATUS

approved

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Last modified April 6 21:24 EDT 2020. Contains 333286 sequences. (Running on oeis4.)