A260428 Composite numbers whose binary representations encode a polynomial (with coefficients 0 or 1) which is irreducible over Q, but reducible over GF(2).

69, 77, 81, 121, 169, 205, 209, 261, 265, 275, 289, 295, 305, 321, 323, 327, 329, 339, 377, 405, 407, 437, 453, 473, 475, 481, 493, 517, 533, 551, 553, 559, 565, 575, 581, 583, 595, 625, 649, 667, 671, 689, 703, 707, 737, 747, 749, 755, 763, 767, 779, 781, 785, 805, 815, 833, 835, 851, 855, 861, 869, 893, 905

Offset: 1

Author

Antti Karttunen, Jul 26 2015

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..11585

Crossrefs

Intersection of A002808 and A260427.

Intersection of A091212 and A206074.

Intersection of A091242 and A206075.

Complement of A257688 in A206074.

Programs

Maple

f:= proc(n) local L,p,x;
  if isprime(n) then return false fi;
  L:= convert(n,base,2);
  p:= add(L[i]*x^(i-1),i=1..nops(L));
  irreduc(p) and not (Irreduc(p) mod 2);
end proc:
select(f, [$2..10000]); # Robert Israel, Jul 27 2015

Mathematica

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

PARI

isA260428(n) = (polisirreducible( Pol(binary(n)) ) && !polisirreducible(Pol(binary(n))*Mod(1, 2)) && !isprime(n));
n = 0; i = 0; while(n < 65537, n++; if(isA260428(n), i++; write("b260428.txt", i, " ", n)));