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%I #38 Feb 28 2016 14:05:22
%S 5,17,23,29,43,53,71,79,83,89,101,107,113,127,139,149,151,163,173,179,
%T 181,197,199,223,227,233,251,257,263,269,271,277,281,293,307,311,317,
%U 331,337,347,349,353,359,367,373,383,389,401,409,421,431,439,443,449,457,461,467,479,491,503,509,521,523
%N Primes whose binary representation encodes a polynomial reducible 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 Except for 3, all primes with even Hamming weight (A027699) are terms, see A238186 for the subsequence of primes with odd Hamming weight. [_Joerg Arndt_ and _Antti Karttunen_, Feb 19 2014]
%H Antti Karttunen, <a href="/A091209/b091209.txt">Table of n, a(n) for n = 1..71800</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(A091210(n)) = A091242(A091211(n)).
%F Other identities. For all n >= 1:
%F A235043(a(n)) = n. [A235043 works as a left inverse of this sequence.]
%p Primes:= select(isprime,[2,seq(2*i+1,i=1..1000)]):
%p filter:= proc(n) local L,x;
%p L:= convert(n,base,2);
%p Irreduc(add(L[i]*x^(i-1),i=1..nops(L))) mod 2;
%p end proc:
%p remove(filter,Primes); # _Robert Israel_, May 17 2015
%t Select[Prime[Range[2, 100]], !IrreduciblePolynomialQ[bb = IntegerDigits[#, 2]; Sum[bb[[k]] x^(k-1), {k, 1, Length[bb]}], Modulus -> 2]&] (* _Jean-François Alcover_, Feb 28 2016 *)
%o (PARI) forprime(p=2, 10^3, if( ! polisirreducible( Mod(1,2)*Pol(binary(p)) ), print1(p,", ") ) ); \\ _Joerg Arndt_, Feb 19 2014
%Y Intersection of A000040 and A091242.
%Y Disjoint union of A238186 and (A027699 \ {3}).
%Y Left inverse: A235043.
%Y Cf. A091206 (Primes whose binary expansion encodes a polynomial irreducible over GF(2)), A091212 (Composite, and reducible over GF(2)), A091214 (Composite, but irreducible over GF(2)).
%Y Cf. also A235041-A235042, A234742.
%K nonn
%O 1,1
%A _Antti Karttunen_, Jan 03 2004
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