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%I #14 Nov 23 2022 10:08:32
%S 1,1,1,3,1,1,1,3,1,1,1,9,1,13,29,3,1,1,37,3,1,23,1,9,49,25,1,39,1,29,
%T 32,93,67,1,71,27,1,37,79,3,83,13,173,69,29,47,1,423,293,49,103,75,
%U 317,53,109,39,37,59,1297,261,367,1024,1,93,1,1541,269,201,277,923,283,1917
%N A divisibility sequence derived from Lehmer's polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1. Square root of the terms in A059928.
%C The sequence is conjectured to contain an infinite number of primes. The first 100 terms contain 33 unique primes. As stated by Everest and Ward, except for a finite number of composite n, a(n) can be prime only if n is prime. For this sequence, n=23*47 is the largest composite for which a(n) is prime.
%D M. Einsiedler, G. Everest, T. Ward, Primes in sequences associated to polynomials, LMS J. Comp. Math. 3 (2000), 15-29
%D G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.
%H G. Everest and T. Ward, <a href="https://ueaeprints.uea.ac.uk/19703/1/cubopdf.pdf">Primes in Divisibility Sequences</a>, Cubo Matematica Educacional (2001), 3 (2), pp. 245-259.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%t 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; Sqrt[Abs[Det[tmp-im]]], {n, 100}]
%Y Cf. A059928.
%K nonn
%O 1,4
%A _T. D. Noe_, Sep 15 2003
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