A091206 Primes whose binary representation encodes a polynomial irreducible over GF(2).

2, 3, 7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 73, 97, 103, 109, 131, 137, 157, 167, 191, 193, 211, 229, 239, 241, 283, 313, 379, 397, 419, 433, 463, 487, 499, 557, 563, 587, 601, 607, 613, 617, 631, 647, 661, 677, 701, 719, 757, 761, 769, 787, 827, 859

Offset

1,1

Comments

"Encoded in binary representation" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where each coefficient a(k) = 0 or 1).

Subsequence with Hamming weight nonprime starts 2, 1019, 1279, 1531, 1663, 1759, 1783, 1789, 2011, 2027, 2543, 2551, ... [Joerg Arndt, Nov 01 2013]. These are now given by A255569. - Antti Karttunen, May 14 2015

Links

Antti Karttunen, Table of n, a(n) for n = 1..10226; all terms up to 1048357, binary length 20

A. Karttunen, Scheme-program for computing this sequence.

Index entries for sequences related to binary encoded polynomials over GF(2)

Formula

a(n) = A000040(A091207(n)) = A014580(A091208(n)).

Mathematica

okQ[p_] := Module[{id, pol, x}, id = IntegerDigits[p, 2] // Reverse; pol = id.x^Range[0, Length[id] - 1]; IrreduciblePolynomialQ[pol, Modulus -> 2]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Feb 06 2023 *)

Prog

(PARI)
is(n)=polisirreducible( Mod(1, 2) * Pol(digits(n, 2)) );
forprime(n=2, 10^3, if (is(n), print1(n, ", ")));
\\ Joerg Arndt, Nov 01 2013

Crossrefs

Intersection of A014580 and A000040.

Apart from a(2) = 3 a subsequence of A027697. The numbers in A027697 but not here are listed in A238186.

Also subsequence of A235045 (its primes. Cf. also A235041-A235042).

Cf. A091209 (Primes whose binary expansion encodes a polynomial reducible over GF(2)), A091212 (Composite, and reducible over GF(2)), A091214 (Composite, but irreducible over GF(2)), A257688 (either 1, prime or irreducible over GF(2)).

Subsequence: A255569.

Sequence in context: A155153 A014580 A197227 * A357713 A038963 A167609

Adjacent sequences: A091203 A091204 A091205 * A091207 A091208 A091209

Keyword

nonn

Author

Antti Karttunen, Jan 03 2004

Status

approved