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 A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2. 103
 2, 3, 7, 11, 13, 19, 25, 31, 37, 41, 47, 55, 59, 61, 67, 73, 87, 91, 97, 103, 109, 115, 117, 131, 137, 143, 145, 157, 167, 171, 185, 191, 193, 203, 211, 213, 229, 239, 241, 247, 253, 283, 285, 299, 301, 313, 319, 333, 351, 355, 357, 361, 369, 375 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10. The numbers {a(n)} are a subset of the set {A206074}. - Thomas Ordowski, Feb 21 2014 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 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 LINKS A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (first 1377 terms from T. D. Noe) EXAMPLE x^4 + x^3 + 1 -> 16+8+1 = 25. Or, x^4 + x^3 + 1 -> 11001 (binary) = 25 (decimal). MATHEMATICA 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 *) 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 *) PROG (PARI) is(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)) \\ Charles R Greathouse IV, Mar 22 2013 CROSSREFS Written in binary: A058943. Number of degree-n irreducible polynomials: A001037, see also A000031. Multiplication table: A048720. 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. Table of irreducible factors of n: A256170. Irreducible polynomials satisfying particular conditions: A071642, A132447, A132449, A132453, A162570. Factorization sentinel: A278239. 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. Factorization-preserving isomorphisms: A091203, A091204, A235041, A235042. See A115871 for sequences related to cross-domain congruences. Functions based on the irreducibles: A305421, A305422. Sequence in context: A321657 A040116 A155153 * A197227 A091206 A038963 Adjacent sequences:  A014577 A014578 A014579 * A014581 A014582 A014583 KEYWORD nonn AUTHOR David Petry (petry(AT)accessone.com) STATUS approved

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Last modified May 23 23:52 EDT 2022. Contains 353993 sequences. (Running on oeis4.)