Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #33 Feb 06 2023 10:02:16
%S 2,3,7,11,13,19,31,37,41,47,59,61,67,73,97,103,109,131,137,157,167,
%T 191,193,211,229,239,241,283,313,379,397,419,433,463,487,499,557,563,
%U 587,601,607,613,617,631,647,661,677,701,719,757,761,769,787,827,859
%N Primes whose binary representation encodes a polynomial irreducible over GF(2).
%C "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).
%C 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
%H Antti Karttunen, <a href="/A091206/b091206.txt">Table of n, a(n) for n = 1..10226; all terms up to 1048357, binary length 20</a>
%H A. Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing this sequence.</a>
%H <a href="/index/Ge#GF2X">Index entries for sequences related to binary encoded polynomials over GF(2)</a>
%F a(n) = A000040(A091207(n)) = A014580(A091208(n)).
%t okQ[p_] := Module[{id, pol, x}, id = IntegerDigits[p, 2] // Reverse; pol = id.x^Range[0, Length[id] - 1]; IrreduciblePolynomialQ[pol, Modulus -> 2]];
%t Select[Prime[Range[1000]], okQ] (* _Jean-François Alcover_, Feb 06 2023 *)
%o (PARI)
%o is(n)=polisirreducible( Mod(1,2) * Pol(digits(n,2)) );
%o forprime(n=2,10^3,if (is(n), print1(n,", ")));
%o \\ _Joerg Arndt_, Nov 01 2013
%Y Intersection of A014580 and A000040.
%Y Apart from a(2) = 3 a subsequence of A027697. The numbers in A027697 but not here are listed in A238186.
%Y Also subsequence of A235045 (its primes. Cf. also A235041-A235042).
%Y 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)).
%Y Subsequence: A255569.
%K nonn
%O 1,1
%A _Antti Karttunen_, Jan 03 2004