A001115 Maximal number of pairwise relatively prime polynomials of degree n over GF(2). (Formerly M0575 N0209)

1, 2, 3, 4, 6, 9, 14, 23, 38, 64, 113, 200, 358, 653, 1202, 2223, 4151, 7781, 14659, 27721, 52603, 100084, 190969, 365134, 699617, 1342923, 2582172, 4972385, 9588933, 18515328, 35794987, 69278386, 134224480, 260309786, 505302925, 981723316, 1908898002, 3714597352, 7233673969, 14096361346, 27487875487

Offset

0,2

Comments

For n>=4, a maximal set can be chosen by taking all irreducible polynomials of degree n, the squares of all irreducible polynomials of degree n/2 (if n is even) and, for each irreducible polynomial p of degree d with 1 <= d < n/2, a product p*q where q is irreducible of degree n-d. The q's should all be distinct, which is possible when n>=4.

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..40.

D. C. Bossen and S. S. Yau, Redundant residue polynomial codes, Information and Control 13 (1968) 597-618.

Formula

a(n) = P(n) + Sum_{i=1..floor(n/2)} P(i), where P(n) = A001037(n) = number of irreducible polynomials of degree n.

Example

n=1: x and x+1.
n=2: x^2, x^2+1, x^2+x+1.
n=3: x^3, x^3+1, x^3+x+1, x^3+x^2+1.

Mathematica

p[0]=1; p[n_] := Sum[If[Mod[n, d]==0, MoebiusMu[n/d]2^d, 0], {d, 1, n}]/n; a[n_] := p[n]+Sum[p[i], {i, 1, Floor[n/2]}]

Prog

(PARI) A001115(n)=A001037(n)+sum(i=1, n\2, A001037(i)) \\ M. F. Hasler, Jan 11 2016

Crossrefs

Cf. A001037.

Sequence in context: A005579 A000381 A350588 * A173278 A173289 A096824

Adjacent sequences: A001112 A001113 A001114 * A001116 A001117 A001118

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Dean Hickerson, Nov 18 2002

More terms from M. F. Hasler, Jan 11 2016

Status

approved