

A059928


The periodic point counting sequence for the toral automorphism given by the polynomial of (conjectural) smallest Mahler measure. The map is x > Ax mod 1 for x in [0,1)^10, where A is the companion matrix for the polynomial x^10+x^9x^7x^6x^5x^4x^3+x+1.


2



1, 1, 1, 9, 1, 1, 1, 9, 1, 1, 1, 81, 1, 169, 841, 9, 1, 1, 1369, 9, 1, 529, 1, 81, 2401, 625, 1, 1521, 1, 841, 1024, 8649, 4489, 1, 5041, 729, 1, 1369, 6241, 9, 6889, 169, 29929, 4761, 841, 2209, 1, 178929, 85849, 2401, 10609, 5625, 100489, 2809, 11881, 1521, 1369
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OFFSET

1,4


COMMENTS

It is expected that the sequence contains infinitely many squares of primes. The heuristics for the Mersenne sequence can be adapted to show that approximately c log N of the first N terms should be prime. The paper Einsiedler, Everest, Ward gives supporting numerical evidence.
The terms in this sequence are all squares. The sequence of square roots, A087612, is conjectured to contain an infinite number of primes.  T. D. Noe, Sep 15 2003


REFERENCES

M. Einsiedler, G. Everest, T. Ward, Primes in sequences associated to polynomials, LMS J. Comp. Math. 3 (2000), 1529
G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.


LINKS

Table of n, a(n) for n=1..57.
G. Everest and T. Ward, Primes in Divisibility Sequences
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index to divisibility sequences


FORMULA

The nth term is abs(det(A^nI)) where I is the 10 by 10 identity matrix and A is the companion matrix to the polynomial x^10+x^9x^7x^6x^5x^4x^3+x+1.


EXAMPLE

The first term is 1 because Ax=x implies x=0 (since AI) is invertible. Thus there is only one fixed point for the map.


MATHEMATICA

CompanionMatrix[p_, x_] := Module[{cl=CoefficientList[p, x], deg, m}, cl=Drop[cl/Last[cl], 1]; deg=Length[cl]; If[deg==1, {cl}, m=RotateLeft[IdentityMatrix[deg]]; m[[ 1]]=cl; Transpose[m]]]; c=CompanionMatrix[x^10+x^9x^7x^6x^5x^4x^3+x+1, x]; im=IdentityMatrix[10]; tmp=im; Table[tmp=tmp.c; Abs[Det[tmpim]], {n, 100}] (From T. D. Noe)


CROSSREFS

Cf. A060478, A087612.
Sequence in context: A110483 A010164 A006084 * A283989 A180839 A239797
Adjacent sequences: A059925 A059926 A059927 * A059929 A059930 A059931


KEYWORD

nonn


AUTHOR

Graham Everest (g.everest(AT)uea.ac.uk), Mar 01 2001


EXTENSIONS

More terms from T. D. Noe, Sep 15 2003


STATUS

approved



