A211672 Least number k such that the polynomial x^n - x^(n-1) -...- 1 (mod k) has more than n distinct zeros.

209, 517, 3973, 1081, 1285, 2893, 13501, 38579, 105113, 4897, 12331

Offset

2,1

Comments

This is the characteristic polynomial of the n-step Fibonacci and Lucas sequences. These terms produce the following number of distinct zeros: 4, 6, 8, 6, 8, 8, 10, 12, 15, 12, 18. The first 11 terms are semiprimes; the 12th term has 3 factors. For prime k, the polynomial can have at most n zeros.

Links

Table of n, a(n) for n=2..12.

Mathematica

Clear[x]; Table[poly = x^n - Sum[x^k, {k, 0, n - 1}]; k = 1; While[cnt = 0; Do[If[Mod[poly, k] == 0, cnt++], {x, 0, k-1}]; cnt <= n, k++]; k, {n, 2, 7}]

Crossrefs

Cf. A211671 (for prime k).

Sequence in context: A337778 A304154 A305507 * A250780 A305173 A316761

Adjacent sequences: A211669 A211670 A211671 * A211673 A211674 A211675

Keyword

nonn,hard,more

Author

T. D. Noe, Apr 19 2012

Status

approved