A171810 Let a(0) = 1. a(n) is the least k>0 such that k*x^n + Sum_{i=0..n-1} a(i)*x^i is an irreducible polynomial.

1, 1, 1, 2, 2, 3, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1

Offset

0,4

Comments

The program given generates the polynomial coefficients beginning with constant term 1. The cross-referenced A171811 gives the gap-length between nonunit entries, and A171812 gives the degrees that hold coefficients other than 1 and 2, only one of which is not 3.

Is a(n) = 1 for almost all n, that is, does a(n) = 1 with probability 1? - Charles R Greathouse IV, Jul 13 2013

Links

Table of n, a(n) for n=0..104.

Example

The sequence of polynomials starts 1, x+1, x^2+x+1, 2x^3+x^2+x+1, 2x^4+2x^3+x^2+x+1, 3x^5+2x^4+2x^3+x^2+x+1, ... .
The value for the x^5 term is determined by the fact that 1 in its place yields factor x+1 and 2 in its place yields factor x^2+x+1.

Mathematica

a[0] = 1; a[n_] := a[n] = Block[{k = 1}, While[! IrreduciblePolynomialQ[k x^n + Sum[a[i] x^i, {i, 0, n - 1}]], k++]; k]; Table[a@ n, {n, 0, 104}] (* Michael De Vlieger, Dec 10 2015 *)

Prog

(PARI) x=1; for(d=1, 1000, x+=v^d; c=1; while(!polisirreducible(x), c++; x+=v^d; next); print1(c", "))

Crossrefs

Cf. A171811, A171812.

Sequence in context: A241815 A051521 A319562 * A330155 A327759 A269978

Adjacent sequences: A171807 A171808 A171809 * A171811 A171812 A171813

Keyword

nonn

Author

James G. Merickel, Dec 19 2009

Status

approved