%I #34 Nov 02 2023 12:33:19

%S 8,13,16,19,24,26,27,32,37,38,40,43,45,48,50,51,53,56,59,61,64,67,69,

%T 70,72,75,77,78,80,82,83,85,88,91,96,99,101,104,107,109,112,114,115,

%U 116,117,120,122,125,128,131,133,136,138,139,141,143,144,149,152,157

%N Numbers k such that x^k + x^m + 1 is factorable over GF(2) for all m between 1 and k.

%C Brent, Hart, Kruppa, and Zimmermann found that 57885161 is a term of this sequence. - _Charles R Greathouse IV_, May 30 2013

%H Charles R Greathouse IV, <a href="/A057486/b057486.txt">Table of n, a(n) for n = 1..5000</a> (first 200 terms from T. D. Noe)

%H Richard Brent, <a href="http://maths-people.anu.edu.au/~brent/gf2x.html">The Software gf2x</a>.

%H Paul Zimmermann, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;ebe12819.1305">There is no primitive trinomial of degree 57885161 over GF(2)</a>, posting to NMBRTHRY mailing list [<a href="http://webloria.loria.fr/~zimmerma/irred/57885161.txt">alternate link</a>]

%H <a href="/index/Tri#trinomial">Index entries for sequences related to trinomials over GF(2)</a>

%e a(1) = 8 because

%e x^8 + x^1 + 1 = (1 + x + x^2)*(1 + x^2 + x^3 + x^5 + x^6),

%e x^8 + x^2 + 1 = (1 + x + x^4)^2,

%e x^8 + x^3 + 1 = (1 + x + x^3)*(1 + x + x^2 + x^3 + x^5),

%e x^8 + x^4 + 1 = (1 + x + x^2)^4,

%e x^8 + x^5 + 1 = (1 + x^2 + x^3)*(1 + x^2 + x^3 + x^4 + x^5),

%e x^8 + x^6 + 1 = (1 + x^3 + x^4)^2, and

%e x^8 + x^7 + 1 = (1 + x + x^2)*(1 + x + x^3 + x^4 + x^6).

%t Do[ k = 1; While[ ToString[ Factor[ x^n + x^k + 1, Modulus -> 2 ]] != ToString[ x^n + x^k + 1 ] && k < n, k++ ]; If[ k == n, Print[ n ]], {n, 2, 234} ]

%o (PARI) is(n)=for(s=1,n\2,if(polisirreducible((x^n+x^s+1)*Mod(1,2)), return(0)));1 \\ _Charles R Greathouse IV_, May 30 2013

%Y Complement of A073571. Cf. A001153, A002475, A073639.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Sep 28 2000