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A107220
Numbers n such that 1 + (x + x^3 + x^5 + x^7 + ...+ x^(2*n+1)) is irreducible over GF(2).
0
1, 3, 5, 7, 9, 13, 23, 27, 31, 37, 63, 69, 117, 119, 173, 219, 223, 247, 307, 363, 383, 495, 695, 987, 2519, 3919, 4633, 6537, 8881, 12841, 15935, 16383, 16519, 26525, 34415, 95139
OFFSET
1,2
COMMENTS
All terms are odd as irreducible polynomials over GF(2) necessarily have an odd number of nonzero coefficients.
Next term > 10^5. - Joerg Arndt, Apr 28 2012
Any subsequent terms are > 321000. - Lucas A. Brown, Dec 02 2022
LINKS
Joerg Arndt, Matters Computational (The Fxtbook), section 40.9.10 "Irreducible alternating polynomials", pp.853
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
EXAMPLE
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).
PROG
(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, ", ")));
(Sage)
p = 1;
P.<x> = GF(2)[]
for n in range(1, 10^5, 2):
p = p + x^(2*(n-1)+1) + x^(2*n+1);
if p.is_irreducible():
print(n)
# Joerg Arndt, Apr 28 2012
CROSSREFS
Sequence in context: A262602 A133847 A134180 * A249412 A098758 A275254
KEYWORD
nonn,hard,more
AUTHOR
Joerg Arndt, Jun 08 2005
EXTENSIONS
More terms from Joerg Arndt, Apr 02 2011 and (terms >=2519), Apr 27 2012
STATUS
approved