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A206074 n-th irreducible polynomial over Q (with coefficients 0 or 1) evaluated at x=2.

%I A206074 #34 Aug 02 2015 17:52:07
%S A206074 2,3,5,7,11,13,17,19,23,25,29,31,37,41,43,47,53,55,59,61,67,69,71,73,
%T A206074 77,79,81,83,87,89,91,97,101,103,107,109,113,115,117,121,127,131,137,
%U A206074 139,143,145,149,151,157,163,167,169,171,173,179,181,185,191,193,197,199,203,205,209,211,213,223,227,229
%N A206074 n-th irreducible polynomial over Q (with coefficients 0 or 1) evaluated at x=2.
%C A206074 Is every prime present?
%C A206074 Yes, see the Filaseta reference. - _Thomas Ordowski_, Feb 19 2014
%C A206074 Corresponding evaluation at x=10 is A206073. - _Michael Somos_, Feb 26 2014
%H A206074 Antti Karttunen, <a href="/A206074/b206074.txt">Table of n, a(n) for n = 1..21692</a>
%H A206074 John Brillhart, Michael Filaseta, Andrew Odlyzko, <a href="http://dx.doi.org/10.4153/CJM-1981-080-0">On an irreducibility theorem of A. Cohn</a>, Canad. J. Math. 33(1981), pp. 1055-1059.
%H A206074 Michael Filaseta, <a href="http://dx.doi.org/10.4153/CJM-1982-097-3">A further generalization of an irreducibility theorem of A. Cohn</a>, Canad J. Math. 34 (1982), pp. 1390-1395.
%F A206074 Other identities and observations. For all n >= 1:
%F A206074 A255574(a(n)) = n.
%e A206074 (See the example at A206073.)
%t A206074 t = Table[IntegerDigits[n, 2], {n, 1, 850}];
%t A206074 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A206074 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A206074 Table[p[n, x], {n, 1, 15}]
%t A206074 u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
%t A206074 AppendTo[u, n]], {n, 300}];
%t A206074 u                          (* A206074 *)
%t A206074 Complement[Range[200], u]  (* A205783 *)
%t A206074 b[n_] := FromDigits[IntegerDigits[u, 2][[n]]]
%t A206074 Table[b[n], {n, 1, 40}]    (* A206073 *)
%o A206074 (PARI) for(n=2, 10^3, if( polisirreducible( Pol(binary(n)) ), print1(n,", ") ) ); \\ _Joerg Arndt_, Feb 19 2014
%Y A206074 Cf. A206073, A205783 (complement), A206075 (nonprime terms), A014580 (irreducible over GF(2), a subsequence of this one), A000040 (primes, also a subsequence), A260427 (terms that are reducible over GF(2)).
%Y A206074 Cf. A255574 (left inverse).
%Y A206074 Cf. also permutations A260421 - A260426.
%Y A206074 Disjoint union of A257688 (without 1) and A260428.
%Y A206074 a(n) differs from A186891(n+1) for the first time at n=21, where a(21) = 67, while A186891(22) = 65, a term missing from here. There are several other sequences that do not diverge until after approx. the twentieth term from this one (see the context-links).
%K A206074 nonn
%O A206074 1,1
%A A206074 _Clark Kimberling_, Feb 03 2012
%E A206074 Clarified name, added more terms, _Joerg Arndt_, Feb 20 2014