A220947 Least prime p such that sum_{k=0}^n F(k+1)*x^{n-k} is irreducible modulo p, where F(j) denotes the Fibonacci number A000045(j).

2, 3, 2, 11, 3, 2, 5, 3, 2, 11, 5, 41, 181, 31, 73, 89, 5, 7, 71, 11, 29, 5, 193, 41, 89, 61, 2, 43, 3, 31, 13, 191, 2, 61, 103, 97, 103, 47, 383, 367, 89, 17, 191, 1627, 193, 163, 5, 337, 349, 23, 149, 193, 199, 233, 173, 617, 593, 59, 113, 151

Offset

1,1

Comments

Conjecture: a(n) <= n^2+12 for all n>0.

Such a phenomenon happens quite often. In fact, for many interesting integer sequences a(k) (k=1,2,3,...), each of the polynomials x^n + sum_{k=0}^n a(k)*x^{n-k} (n>0) is irreducible modulo some prime not exceeding a*n^2+b*n+c, where a, b, c are suitable nonnegative constants.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..450

Example

a(2) = 3 since x^2+x+2 is irreducible modulo 3 but reducible modulo 2.
Note also that a(13) = 181 = 13^2+12.

Mathematica

A[n_, x_]:=A[n, x]=Sum[Fibonacci[k+1]*x^(n-k), {k, 0, n}]
Do[Do[If[IrreduciblePolynomialQ[A[n, x], Modulus->Prime[k]]==True, Print[n, " ", Prime[k]]; Goto[aa]], {k, 1, PrimePi[n^2+12]}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 100}]

Crossrefs

Cf. A000040, A000045, A224416, A224417, A224418, A224480, A224210, A220072, A223934, A218465.

Sequence in context: A247170 A241055 A224418 * A224417 A225787 A012960

Adjacent sequences: A220944 A220945 A220946 * A220948 A220949 A220950

Keyword

nonn

Author

Zhi-Wei Sun, Apr 07 2013

Status

approved