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

2, 1019, 1279, 1531, 1663, 1759, 1783, 1789, 2011, 2027, 2543, 2551, 2687, 2879, 2927, 2999, 3037, 3319, 3517, 3547, 3559, 3709, 3833, 3947, 4007, 4013, 4021, 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

Offset: 1

Author

Antti Karttunen, May 14 2015 after Joerg Arndt's Nov 01 2013 comment in A091206

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A091206 and A084345.

Intersection of A014580 and A255564.

Programs

Maple

filter:= proc(n)
  local a, i,x;
  if not isprime(n) then return false fi;
  a:= convert(n,base,2);
  not isprime(convert(a,`+`)) and (Irreduc(add(x^(i-1)*a[i],i=1..nops(a))) mod 2)
end proc:
select(filter, [2,2*j+1$j=1..10000]); # Robert Israel, May 14 2015

Mathematica

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]]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Feb 09 2023 *)

PARI

isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i = 0; forprime(n=2, 2^31, if(isA014580(n)&&!isprime(hammingweight(n)), i++; write("b255569.txt", i, " ", n); if(i>=10000,return(n))));