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A255569 Primes whose binary representation encodes an irreducible polynomial over GF(2) and has a nonprime number of 1's.

%I A255569 #18 Feb 09 2023 09:58:27
%S A255569 2,1019,1279,1531,1663,1759,1783,1789,2011,2027,2543,2551,2687,2879,
%T A255569 2927,2999,3037,3319,3517,3547,3559,3709,3833,3947,4007,4013,4021,
%U A255569 4051,4073,4591,5023,5039,5051,5107,5563,5591,5743,5821,5981,6067,6271,6607,6637,6779,6959,7079,7351,7411,7517,7541,7591,7603,7727,7741,7823,7907,7963,7993
%N A255569 Primes whose binary representation encodes an irreducible polynomial over GF(2) and has a nonprime number of 1's.
%H A255569 Antti Karttunen, <a href="/A255569/b255569.txt">Table of n, a(n) for n = 1..10000</a>
%p A255569 filter:= proc(n)
%p A255569   local a, i,x;
%p A255569   if not isprime(n) then return false fi;
%p A255569   a:= convert(n,base,2);
%p A255569   not isprime(convert(a,`+`)) and (Irreduc(add(x^(i-1)*a[i],i=1..nops(a))) mod 2)
%p A255569 end proc:
%p A255569 select(filter, [2,2*j+1$j=1..10000]); # _Robert Israel_, May 14 2015
%t A255569 okQ[p_?PrimeQ] := Module[{id, pol, x}, id = IntegerDigits[p, 2] // Reverse; pol = id.x^Range[0, Length[id]-1]; IrreduciblePolynomialQ[pol, Modulus -> 2] && !PrimeQ[Count[id, 1]]];
%t A255569 Select[Prime[Range[1000]], okQ] (* _Jean-François Alcover_, Feb 09 2023 *)
%o A255569 (PARI)
%o A255569 isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from _Charles R Greathouse IV_
%o A255569 i = 0; forprime(n=2, 2^31, if(isA014580(n)&&!isprime(hammingweight(n)), i++; write("b255569.txt", i, " ", n); if(i>=10000,return(n))));
%Y A255569 Intersection of A091206 and A084345.
%Y A255569 Intersection of A014580 and A255564.
%K A255569 nonn
%O A255569 1,1
%A A255569 _Antti Karttunen_, May 14 2015 after _Joerg Arndt_'s Nov 01 2013 comment in A091206