login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A058943 Coefficients of irreducible polynomials over GF(2) listed in lexicographic order. 22
10, 11, 111, 1011, 1101, 10011, 11001, 11111, 100101, 101001, 101111, 110111, 111011, 111101, 1000011, 1001001, 1010111, 1011011, 1100001, 1100111, 1101101, 1110011, 1110101, 10000011, 10001001, 10001111, 10010001 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Church's table extends through degree 11.

REFERENCES

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983, Table C, pp. 553-555.

LINKS

T. D. Noe, Table of n, a(n) for n=1..1377 (through degree 13)

R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209.

F. Ruskey, Irreducible and Primitive Polynomials over GF(2)

Index entries for sequences containing GF(2)[X]-polynomials

EXAMPLE

The first few are x, x+1; x^2+x+1; x^3+x+1, x^3+x^2+1; ... Note that x is irreducible but not primitive.

MATHEMATICA

Do[a = Reverse[ IntegerDigits[n, 2]]; b = {0}; l = Length[a]; k = 1; While[k < l + 1, b = Append[b, a[[k]]*x^(k - 1) ]; k++ ]; b = Apply[Plus, b]; c = Factor[b, Modulus -> 2]; If[b == c, Print[ FromDigits[ IntegerDigits[n, 2]]]], {n, 3, 250, 2} ]

PROG

(PARI)

seq(N, p=2, maxdeg=oo) = {

  my(a = List(), k=0, d=0);

  while (d++ <= maxdeg,

    for (n=p^d, 2*p^d-1, my(f=Mod(Pol(digits(n, p)), p));

      if(polisirreducible(f), listput(a, subst(lift(f), 'x, 10)); k++);

      if(k >= N, break(2))));

  Vec(a);

};

seq(27) \\ Gheorghe Coserea, May 28 2018

CROSSREFS

Cf. A000020, A001037, A011260, A058944-A058948.

Converted to decimal: A014580.

Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): this sequence, A058944, A058948, A058945, A058946. Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.

Sequence in context: A287948 A287626 A059458 * A222473 A041217 A041218

Adjacent sequences:  A058940 A058941 A058942 * A058944 A058945 A058946

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Jan 13 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 15 21:06 EDT 2018. Contains 316237 sequences. (Running on oeis4.)