A206074 n-th irreducible polynomial over Q (with coefficients 0 or 1) evaluated at x=2.

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 115, 117, 121, 127, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 171, 173, 179, 181, 185, 191, 193, 197, 199, 203, 205, 209, 211, 213, 223, 227, 229

Offset: 1

Author

Clark Kimberling, Feb 03 2012

Keywords

• nonn

Comments

Is every prime present?

Yes, see the Filaseta reference. - Thomas Ordowski, Feb 19 2014

Corresponding evaluation at x=10 is A206073. - Michael Somos, Feb 26 2014

Examples

			(See the example at A206073.)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..21692
• John Brillhart, Michael Filaseta, Andrew Odlyzko, On an irreducibility theorem of A. Cohn, Canad. J. Math. 33(1981), pp. 1055-1059.
• Michael Filaseta, A further generalization of an irreducibility theorem of A. Cohn, Canad J. Math. 34 (1982), pp. 1390-1395.

Crossrefs

Cf. A206073, A205783 (complement), A206075 (nonprime terms), A014580 (irreducible over GF(2), a subsequence of this one), A000040 (primes, also a subsequence), A260427 (terms that are reducible over GF(2)).

Cf. A255574 (left inverse).

Cf. also permutations A260421 - A260426.

Disjoint union of A257688 (without 1) and A260428.

a(n) differs from A186891(n+1) for the first time at n=21, where a(21) = 67, while A186891(22) = 65, a term missing from here. There are several other sequences that do not diverge until after approx. the twentieth term from this one (see the context-links).

Programs

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 850}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 15}]
u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
AppendTo[u, n]], {n, 300}];
u                          (* A206074 *)
Complement[Range[200], u]  (* A205783 *)
b[n_] := FromDigits[IntegerDigits[u, 2][[n]]]
Table[b[n], {n, 1, 40}]    (* A206073 *)

PARI

for(n=2, 10^3, if( polisirreducible( Pol(binary(n)) ), print1(n,", ") ) ); \\ Joerg Arndt, Feb 19 2014

Formula

Other identities and observations. For all n >= 1:

A255574(a(n)) = n.

Extensions

Clarified name, added more terms, Joerg Arndt, Feb 20 2014