A211671 -id:A211671 - cp's OEIS frontend

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

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T. D. Noe, Apr 19 2012

nonn

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.

Cf. A211671 (for prime k).

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}]

More terms from Jinyuan Wang, Apr 25 2025

cp's OEIS Frontend

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

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