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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.

%I A059928 #21 Nov 23 2022 10:08:22
%S A059928 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,
%T A059928 1,1521,1,841,1024,8649,4489,1,5041,729,1,1369,6241,9,6889,169,29929,
%U A059928 4761,841,2209,1,178929,85849,2401,10609,5625,100489,2809,11881,1521,1369
%N 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.
%C A059928 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.
%C A059928 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
%D A059928 G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.
%H A059928 Manfred Einsiedler, Graham Everest and Thomas Ward, <a href="https://doi.org/10.1112/S1461157000000255">Primes in sequences associated to polynomials (after Lehmer)</a>, LMS J. Comput. Math. 3 (2000), 125-139.
%H A059928 G. Everest and T. Ward, <a href="https://ueaeprints.uea.ac.uk/id/eprint/19703">Primes in Divisibility Sequences</a>, Cubo Matematica Educacional (2001), 3 (2), pp. 245-259.
%H A059928 Y. Puri and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A059928 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A059928 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.
%e A059928 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.
%t A059928 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)
%o A059928 (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)));
%o A059928 id(nn) = matrix(nn, nn, n, k, n==k);
%o A059928 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
%Y A059928 Cf. A060478, A087612.
%K A059928 nonn
%O A059928 1,4
%A A059928 Graham Everest (g.everest(AT)uea.ac.uk), Mar 01 2001
%E A059928 More terms from _T. D. Noe_, Sep 15 2003