A107220 - OEIS

%I #39 Dec 02 2022 10:56:14

%S 1,3,5,7,9,13,23,27,31,37,63,69,117,119,173,219,223,247,307,363,383,

%T 495,695,987,2519,3919,4633,6537,8881,12841,15935,16383,16519,26525,

%U 34415,95139

%N Numbers n such that 1 + (x + x^3 + x^5 + x^7 + ...+ x^(2*n+1)) is irreducible over GF(2).

%C All terms are odd as irreducible polynomials over GF(2) necessarily have an odd number of nonzero coefficients.

%C Next term > 10^5. - _Joerg Arndt_, Apr 28 2012

%C Any subsequent terms are > 321000. - _Lucas A. Brown_, Dec 02 2022

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 40.9.10 "Irreducible alternating polynomials", pp.853

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%e The number 5 is in the sequence because x^11 + x^9 + x^7 + x^5 + x^3 + x + 1 is irreducible over GF(2) (and 11 = 2*5 + 1).

%o (PARI) forstep(d=1,10^5,2, p=(1+sum(t=0,d,x^(2*t+1))); if(polisirreducible(Mod(1,2)*p), print1(d,", ")));

%o (Sage)

%o p = 1;

%o P.<x> = GF(2)[]

%o for n in range(1, 10^5, 2):

%o     p = p + x^(2*(n-1)+1) + x^(2*n+1);

%o     if p.is_irreducible():

%o         print(n)

%o # _Joerg Arndt_, Apr 28 2012

%K nonn,hard,more

%O 1,2

%A _Joerg Arndt_, Jun 08 2005

%E More terms from _Joerg Arndt_, Apr 02 2011 and (terms >=2519), Apr 27 2012