A059928 -id:A059928 - cp's OEIS frontend

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

T. D. Noe, Sep 15 2003

nonn

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.

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.

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

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

A060478 Number of orbits of length n in map whose periodic points are A059928.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 12, 56, 0, 0, 0, 72, 0, 0, 24, 0, 0, 96, 24, 0, 48, 0, 0, 33, 270, 136, 0, 144, 18, 0, 0, 160, 0, 168, 0, 696, 96, 0, 48, 0, 3726, 1752, 0, 208, 96, 1896, 52, 216, 0, 0, 60, 28512, 1120, 2208, 16896, 0, 0, 0, 35904, 1080, 594, 1112, 12096

Offset: 1

Thomas Ward

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

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.

Cf. A059928.

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);
b(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)))); \\ A059928
a(n) = sumdiv(n, d, moebius(d)*b(n/d))/n; \\ Michel Marcus, Nov 23 2022

a(n) = (1/n) * Sum_{ d divides n } mu(d) * A059928(n/d).

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

cp's OEIS Frontend

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.

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