A087612 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.

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, 32, 93, 67, 1, 71, 27, 1, 37, 79, 3, 83, 13, 173, 69, 29, 47, 1, 423, 293, 49, 103, 75, 317, 53, 109, 39, 37, 59, 1297, 261, 367, 1024, 1, 93, 1, 1541, 269, 201, 277, 923, 283, 1917

Offset

1,4

Comments

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.

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..72.

G. Everest and T. Ward, Primes in Divisibility Sequences, Cubo Matematica Educacional (2001), 3 (2), pp. 245-259.

Index to divisibility sequences

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; Sqrt[Abs[Det[tmp-im]]], {n, 100}]

Crossrefs

Cf. A059928.

Sequence in context: A306346 A060901 A351545 * A260626 A155828 A226203

Adjacent sequences: A087609 A087610 A087611 * A087613 A087614 A087615

Keyword

nonn

Author

T. D. Noe, Sep 15 2003

Status

approved