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%I #30 May 06 2022 13:13:51
%S 1,3,3,3,5,3,3,29,17,9,5,83,27,43,3,45,9,39,39,9,5,3,33,27,9,71,39,9,
%T 5,83,9,175,83,231,5,119,63,99,17,57,9,63,89,101,27,303,33,183,113,29,
%U 75,9,71,125,71,149,45,99,123,3,39,105,3,27,27,365,39,163
%N First primitive polynomial over GF(2) of degree n, X^n suppressed.
%C 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.
%H Francois R. Grieu, <a href="/A132448/b132448.txt">Table of n, a(n) for n = 1..400</a> (using Joerg Arndt's table).
%H Joerg Arndt, <a href="https://www.jjj.de/mathdata/lowbit-primpoly.txt">Binary primitive polynomials with lowest-most possible set bits</a>
%H <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>
%e 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.
%t 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 *)
%Y 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.
%K nonn
%O 1,2
%A _Francois R. Grieu_, Aug 22 2007
%E More terms from _Francois R. Grieu_, Jan 12 2021
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