A107220 Numbers k such that 1 + (x + x^3 + x^5 + x^7 + ... + x^(2*k+1)) is irreducible over GF(2).

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, 502147, 524163, 619659, 663781, 806183

Offset

1,2

Comments

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

a(42) (if it exists) is greater than 10^6. - Robin Visser, Oct 11 2025

Links

Table of n, a(n) for n=1..41.

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, ", ")));
(SageMath)
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

Cf. A071428, A071522, A071642.

Sequence in context: A262602 A133847 A134180 * A249412 A098758 A275254

Adjacent sequences: A107217 A107218 A107219 * A107221 A107222 A107223

Keyword

nonn,hard,more

Author

Joerg Arndt, Jun 08 2005

Extensions

More terms from Joerg Arndt, Apr 02 2011 and terms a(25)-a(36), Apr 27 2012

a(37)-a(41) from Robin Visser, Oct 11 2025

Status

approved