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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)

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]] (* Jean-Fran├žois 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 October 27 10:47 EDT 2021. Contains 348274 sequences. (Running on oeis4.)