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A057463 Numbers k such that x^k + x^4 + 1 is irreducible over GF(2).

%I A057463 #42 Nov 29 2022 01:19:10
%S A057463 1,3,7,9,15,39,57,81,105,1239,5569,9457,11095,11631,12327,37633,63247,
%T A057463 216457
%N A057463 Numbers k such that x^k + x^4 + 1 is irreducible over GF(2).
%C A057463 a(18) is greater than 10^5. - _Joerg Arndt_, Apr 28 2012
%C A057463 All terms are congruent to 1 or 3 (mod 6). - _Robert Israel_, Sep 05 2016
%C A057463 Any subsequent terms are > 300000. - _Lucas A. Brown_, Nov 28 2022
%H A057463 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p.850
%H A057463 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/irred_trinom_f2.py">Python program</a>.
%H A057463 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/irred_trinom_f2.sage">Sage program</a>.
%e A057463 6 is not in the sequence since x^6 + x^4 + 1 = (x^3 + x^2 + 1)^2, but 7 is in the sequence since x^7 + x^4 + 1 is irreducible. (Trial division by x + 1, x^2 + x + 1, x^3 + x^2 + 1, and x^3 + x + 1) - _Michael B. Porter_, Sep 06 2016
%p A057463 for m from 1 to 200 do if(Irreduc(x^m + x^4 + 1) mod 2) then printf("%d, ",m):fi:od: # _Nathaniel Johnston_, Apr 19 2011
%o A057463 (Sage)
%o A057463 P.<x> = GF(2)[]
%o A057463 for n in range(10^4):
%o A057463     if (x^n+x^4+1).is_irreducible():
%o A057463         print(n) # _Joerg Arndt_, Apr 28 2012
%Y A057463 Cf. A002475.
%K A057463 nonn,more
%O A057463 1,2
%A A057463 _Robert G. Wilson v_, Sep 27 2000
%E A057463 a(10)-a(15) from _Nathaniel Johnston_, Apr 19 2011
%E A057463 a(16)-a(17) from _Joerg Arndt_, Apr 28 2012
%E A057463 a(18) from _Lucas A. Brown_, Nov 28 2022