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

Author

Francois R. Grieu, Aug 22 2007

Keywords

• nonn

Examples

			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.
		

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

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.

Programs

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 *)