A132447 First primitive GF(2)[X] polynomial of degree n.

3, 7, 11, 19, 37, 67, 131, 285, 529, 1033, 2053, 4179, 8219, 16427, 32771, 65581, 131081, 262183, 524327, 1048585, 2097157, 4194307, 8388641, 16777243, 33554441, 67108935, 134217767, 268435465, 536870917, 1073741907, 2147483657

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..330

Rich Schroeppel, Hasty Pudding Cipher Specification on archive.org, (revised May 1999 ed.), June 1998. (The numbers are called "Swizpoly numbers" here, except that they start with 0 for some reason.)

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

Example

a(5)=37, or 100101 in binary, representing the GF(2)[X] polynomial X^5+X^2+1, because it has degree 5 and is primitive, contrary to X^5, X^5+1, X^5+x^1, X^5+X^1+1 and X^5+X^2.

Maple

f:= proc(n) local k, L, i, X;
   for k from 2^n+1 by 2 do
     L:= convert(k, base, 2);
     if Primitive(add(L[i]*X^(i-1), i=1..n+1)) mod 2 then return k fi
   od
end proc:
map(f, [$1..40]); # Robert Israel, Nov 05 2023

Mathematica

f[n_] := If[n == 1, 3, Module[{k, L, i, X}, For[k = 2^n+1, True, k = k+2, L = IntegerDigits[k, 2]; If[PrimitivePolynomialQ[Sum[L[[i]]*X^(i-1), {i, 1, n+1}], 2], Return[k]]]]];
Table[f[n], {n, 1, 40}] (* Jean-François Alcover, Mar 29 2024, after Robert Israel *)

Crossrefs

a(n) is the smallest member of A091250 at least 2^n. A132448(n) = a(n)-2^n, giving a more compact representation. Cf. A132449, similar, with restriction to at most 5 terms. Cf. A132451, similar, with restriction to exactly 5 terms. Cf. A132453, similar, with restriction to minimal number of terms.

Sequence in context: A354902 A022406 A355288 * A132449 A132453 A033871

Adjacent sequences: A132444 A132445 A132446 * A132448 A132449 A132450

Keyword

nonn

Author

Francois R. Grieu, Aug 22 2007

Status

approved