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%I #23 Feb 06 2023 05:02:20
%S 69,77,81,121,169,205,209,261,265,275,289,295,305,321,323,327,329,339,
%T 377,405,407,437,453,473,475,481,493,517,533,551,553,559,565,575,581,
%U 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
%N Composite numbers whose binary representations encode a polynomial (with coefficients 0 or 1) which is irreducible over Q, but reducible over GF(2).
%H Antti Karttunen, <a href="/A260428/b260428.txt">Table of n, a(n) for n = 1..11585</a>
%p f:= proc(n) local L,p,x;
%p if isprime(n) then return false fi;
%p L:= convert(n,base,2);
%p p:= add(L[i]*x^(i-1),i=1..nops(L));
%p irreduc(p) and not (Irreduc(p) mod 2);
%p end proc:
%p select(f, [$2..10000]); # _Robert Israel_, Jul 27 2015
%t 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]];
%t Select[Range[1000], okQ] (* _Jean-François Alcover_, Feb 06 2023 *)
%o (PARI)
%o isA260428(n) = (polisirreducible( Pol(binary(n)) ) && !polisirreducible(Pol(binary(n))*Mod(1, 2)) && !isprime(n));
%o n = 0; i = 0; while(n < 65537, n++; if(isA260428(n), i++; write("b260428.txt", i, " ", n)));
%Y Intersection of A002808 and A260427.
%Y Intersection of A091212 and A206074.
%Y Intersection of A091242 and A206075.
%Y Complement of A257688 in A206074.
%K nonn
%O 1,1
%A _Antti Karttunen_, Jul 26 2015
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