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, 262999, 18659, 131887, 129373, 109901, 149477, 1438121, 391229, 4244563, 160853, 196031, 5187263, 946679, 1312145, 507727, 870017, 1577593, 234973, 572977, 991349, 2279233, 1476029, 1451299

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..35.

Mathematica

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

Author

T. D. Noe, Apr 19 2012

Extensions

More terms from Jinyuan Wang, Apr 25 2025

Status

approved