login
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.
2
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
G. Everest and T. Ward, Primes in Divisibility Sequences, Cubo Matematica Educacional (2001), 3 (2), pp. 245-259.
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
KEYWORD
nonn
AUTHOR
T. D. Noe, Sep 15 2003
STATUS
approved