

A128976


Number of cycles for the map LL:x>x^22 acting on Z/(2^n1).


2



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

0,1


COMMENTS

A cycle is the orbit of an element x of Z/(2^n1) 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) 219240.


FORMULA

If p=2^n1 is prime, then a(n) = 1/2 + sum_{d2^(n1)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^q1, 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+((i1)^22)%Mq], v[i]=c++); if(v[i]>start, cyc++)); cyc } A128976=vector(20, i, numcycles(i1))


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



