%I #49 May 11 2021 10:57:22
%S 2,3,7,11,13,19,25,31,37,41,47,55,59,61,67,73,87,91,97,103,109,115,
%T 117,131,137,143,145,157,167,171,185,191,193,203,211,213,229,239,241,
%U 247,253,283,285,299,301,313,319,333,351,355,357,361,369,375
%N Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.
%C Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10.
%C The numbers {a(n)} are a subset of the set {A206074}. - _Thomas Ordowski_, Feb 21 2014
%C 2^n - 1 is a term if and only if n = 2 or n is a prime and 2 is a primitive root modulo n. - _Jianing Song_, May 10 2021
%C For odd k, k is a term if and only if binary_reverse(k) = A145341((k+1)/2) is. - _Joerg Arndt_ and _Jianing Song_, May 10 2021
%H A.H.M. Smeets, <a href="/A014580/b014580.txt">Table of n, a(n) for n = 1..20000</a> (first 1377 terms from T. D. Noe)
%H <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>
%e x^4 + x^3 + 1 -> 16+8+1 = 25. Or, x^4 + x^3 + 1 -> 11001 (binary) = 25 (decimal).
%t fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus -> 2] && n != 2^Floor@ Log2@ n]; fQ[2] = True; Select[ Range@ 378, fQ] (* _Robert G. Wilson v_, Aug 12 2011 *)
%t Reap[Do[If[IrreduciblePolynomialQ[IntegerDigits[n, 2] . x^Reverse[Range[0, Floor[Log[2, n]]]], Modulus -> 2], Sow[n]], {n, 2, 1000}]][[2, 1]] (* _Jean-François Alcover_, Nov 21 2016 *)
%o (PARI) is(n)=polisirreducible(Pol(binary(n))*Mod(1,2)) \\ _Charles R Greathouse IV_, Mar 22 2013
%Y Written in binary: A058943.
%Y Number of degree-n irreducible polynomials: A001037, see also A000031.
%Y Multiplication table: A048720.
%Y Characteristic function: A091225. Inverse: A091227. a(n) = A091202(A000040(n)). Almost complement of A091242. Union of A091206 & A091214 and also of A091250 & A091252. First differences: A091223. Apart from a(1) and a(2), a subsequence of A092246 and hence A000069.
%Y Table of irreducible factors of n: A256170.
%Y Irreducible polynomials satisfying particular conditions: A071642, A132447, A132449, A132453, A162570.
%Y Factorization sentinel: A278239.
%Y Sequences analyzing the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the corresponding integer: A234741, A234742, A235032, A235033, A235034, A235035, A235040, A236850, A325386, A325559, A325560, A325563, A325641, A325642, A325643.
%Y Factorization-preserving isomorphisms: A091203, A091204, A235041, A235042.
%Y See A115871 for sequences related to cross-domain congruences.
%Y Functions based on the irreducibles: A305421, A305422.
%K nonn
%O 1,1
%A David Petry (petry(AT)accessone.com)