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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^9-x^7-x^6-x^5-x^4-x^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 (list; graph; refs; listen; history; text; internal format)
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), 15-29

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

EXAMPLE

The first term is 1 because Ax=x implies x=0 (since A-I) 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^9-x^7-x^6-x^5-x^4-x^3+x+1, x]; im=IdentityMatrix[10]; tmp=im; Table[tmp=tmp.c; Abs[Det[tmp-im]], {n, 100}] (From T. D. Noe)

CROSSREFS

Cf. A060478, A087612.

Sequence in context: A110483 A010164 A006084 * A180839 A239797 A010163

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

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Last modified November 1 07:47 EDT 2014. Contains 248888 sequences.