A220949 Least prime p such that sum_{k=0}^n (2k+1)*x^(n-k) is irreducible modulo p.

2, 2, 3, 2, 5, 3, 71, 23, 11, 2, 5, 2, 13, 23, 47, 47, 269, 2, 7, 19, 53, 101, 7, 53, 113, 11, 23, 2, 43, 347, 53, 283, 191, 17, 41, 2, 239, 677, 3, 281, 37, 641, 613, 41, 17, 269, 181, 137, 383, 41, 127, 2, 71, 739, 71, 353, 59, 2, 83, 2

Offset

1,1

Comments

Conjecture: a(n) <= n^2+22 for all n>0.

We have similar conjectures with 2k+1 in the definition replaced by (2k+1)^m (m=2,3,...).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..500

Zhi-Wei Sun, A family of polynomials and a related conjecture on primes, a message to Number Theory List, March 30, 2013.

Example

a(5) = 5 since f(x) = x^5+3*x^4+5*x^3+7*x^2+9*x+11 is irreducible modulo 5, but f(x)==(x+1)*(x^2+x+1)^2 (mod 2) and f(x)==(x+1)^4*(x-1) (mod 3).
Note also that a(7) = 71 = 7^2+22.

Mathematica

A[n_, x_] := A[n, x] = Sum[(2k+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+22]}]; Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 100}]

Crossrefs

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

Sequence in context: A256267 A307687 A281121 * A029656 A121306 A073311

Adjacent sequences: A220946 A220947 A220948 * A220950 A220951 A220952

Keyword

nonn

Author

Zhi-Wei Sun, Apr 07 2013

Status

approved