A057496 Numbers n such that x^n + x^3 + x^2 + x + 1 is irreducible over GF(2).

1, 2, 3, 4, 5, 7, 10, 17, 20, 25, 28, 31, 41, 52, 130, 151, 196, 503, 650, 761, 986, 1391, 2047, 6172, 6431, 6730, 8425, 10162, 11410, 12071, 13151, 14636, 17377, 18023, 32770, 77047, 102842, 130777, 137113, 143503, 168812, 192076

Offset: 1

Author

Robert G. Wilson v, Sep 27 2000

Keywords

• nonn

Comments

If x^n + x^3 + x^2 + x + 1 is irreducible, then so is its "twin" x^n + x^3 + 1. - Gove Effinger, Mar 11 2007

No term other than 3 can be a multiple of 3, since for m > 1, x^(3*m) + x^3 + x^2 + x + 1 is divisible by x^2 + x + 1. - Jianing Song, May 11 2021

Crossrefs

Cf. A002475.

Other than the term 3, subsequence of A057461.

Formula

Using probabilistic arguments it appears that there should be about 6.5 terms in this sequence with any given number of decimal digits d. - Gove Effinger, Mar 11 2007

Extensions

a(20)-a(23) from Robert G. Wilson v, Mar 11 2007

a(24)-a(27) computed by Richard P. Brent, Mar 11 2007, communicated by Gove Effinger

a(27)-a(35) computed by Richard P. Brent, Mar 16 2007, communicated by Gove Effinger

a(36)-a(42) computed by Jonathan Webster, Feb 18 2010

Added entries a(1), a(2), a(3) since x^3 + x^2 + 1, x^3 + x + 1 and x^2 + x + 1 are irreducible over GF(2). Changed the offset for the entries computed by Robert G. Wilson v and Richard P. Brent to account for this. Added terms a(36) through a(42). - Jonathan Webster (jwebster(AT)bates.edu), Feb 18 2010