

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
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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)
Index entries for sequences operating on GF(2)[X]polynomials


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]] (* JeanFrançois 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 degreen 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.
Factorizationpreserving isomorphisms: A091203, A091204, A235041, A235042.
See A115871 for sequences related to crossdomain 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



