OFFSET
1,2
COMMENTS
More precisely: minimum value for X=2 of polynomials P[X] with coefficients in GF(2) such that X^n+P[X] is primitive. Applications include maximum-length linear feedback shift registers with efficient implementation in software.
LINKS
Francois R. Grieu, Table of n, a(n) for n = 1..400 (using Joerg Arndt's table).
EXAMPLE
a(11)=5, or 101 in binary, representing the GF(2)[X] polynomial X^2+1, because X^11+X^2+1 is primitive, contrary to X^11, X^11+1, X^11+X^1, X^11+X^1+1 and X^11+X^2.
MATHEMATICA
i2px[i_]:=If[i>1, BitAnd[i, 1]+i2px[BitShiftRight[i, 1]]x, i ]; s={1}; For[n=2, n<69, ++n, For[i=3, !PrimitivePolynomialQ[i2px[i]+x^n, 2], i+=2]; AppendTo[s, i]]; s (* Francois R. Grieu, Jan 15 2021 *)
CROSSREFS
2^n+a(n) is the smallest member of A091250 at least 2^n. A132447(n) = a(n)+2^n and gives the corresponding primitive polynomial. Cf. A132450, similar, with restriction to at most 5 terms. Cf. A132452, similar, with restriction to exactly 5 terms. Cf. A132454, similar, with restriction to minimal number of terms.
KEYWORD
nonn
AUTHOR
Francois R. Grieu, Aug 22 2007
EXTENSIONS
More terms from Francois R. Grieu, Jan 12 2021
STATUS
approved