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.

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

Offset: 1

Author

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

Keywords

• nonn

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

Examples

			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.
		

References

• G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.

Links

• Manfred Einsiedler, Graham Everest and Thomas Ward, Primes in sequences associated to polynomials (after Lehmer), LMS J. Comput. Math. 3 (2000), 125-139.
• G. Everest and T. Ward, Primes in Divisibility Sequences, Cubo Matematica Educacional (2001), 3 (2), pp. 245-259.
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Index to divisibility sequences

Crossrefs

Cf. A060478, A087612.

Programs

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)

PARI

comp(pol) = my(v=Vec(pol), nn=poldegree(pol)); matrix(nn, nn, n, k, if (k==nn, -v[n], if(k==n-1, 1)));
id(nn) = matrix(nn, nn, n, k, n==k);
a(n) = my(p=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, m=comp(p)); abs(matdet(m^n-id(poldegree(p)))); \\ Michel Marcus, Nov 23 2022

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.

Extensions

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