A062692 Number of irreducible polynomials over F_2 of degree at most n.

2, 3, 5, 8, 14, 23, 41, 71, 127, 226, 412, 747, 1377, 2538, 4720, 8800, 16510, 31042, 58636, 111013, 210871, 401428, 766150, 1465020, 2807196, 5387991, 10358999, 19945394, 38458184, 74248451, 143522117, 277737797, 538038783, 1043325198

Offset

1,1

Comments

Number of binary pre-necklaces of length n. - Joerg Arndt, Jul 20 2013

Links

G. C. Greubel, Table of n, a(n) for n = 1..3320

P. Burcsi, G. Fici, Z. Lipták, F. Ruskey, and J. Sawada, On prefix normal words and prefix normal forms, Preprint, 2016.

G. Fici and Zs. Lipták, On Prefix Normal Words.

G. Fici and Zs. Lipták, On Prefix Normal Words, Developments in Language Theory 2011, Lecture Notes in Computer Science 6795, 228-238.

Kenneth H. Hicks, Gary L. Mullen, and Ikuro Sato, Distribution of irreducible polynomials over F_2, in Finite Fields with Applications to Coding Theory, Cryptography and Related Areas (Oaxaca, 2001), 177-186, Springer, Berlin, 2002.

M. Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.

Formula

a(n) = Sum_{m=1..n} (1/m)*Sum_{d | m } mu(d)*2^{m/d}.

a(n) = A091226(2^(n+1)).

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k)*log(1/(1 - 2*x^k))/k. - Ilya Gutkovskiy, Nov 11 2019

Maple

with(numtheory):for n from 1 to 113 do sum3 := 0:for m from 1 to n do sum2 := 0:a := divisors(m):for h from 1 to nops(a) do sum2 := sum2+mobius(a[h])*2^(m/a[h]):end do:sum3 := sum3+sum2/m:end do:s[n] := sum3:end do:q := seq(s[j], j=1..113);

Mathematica

a[n_] := Sum[1/m DivisorSum[m, MoebiusMu[#]*2^(m/#)&], {m, 1, n}]; Array[a, 34] (* Jean-François Alcover, Dec 07 2015 *)
f[n_] := DivisorSum[n, MoebiusMu[#] * 2^(n/#) &] / n; Accumulate[Array[f, 30]] (* Amiram Eldar, Aug 24 2023 *)

Prog

(PARI) a(n)=sum(m=1, n, 1/m* sumdiv(m, d, moebius(d)*2^(m/d) ) ); /* Joerg Arndt, Jul 04 2011 */

Crossrefs

Partial sums of A001037.

Cf. A014580, A091226, A091231.

Equals A001036 + 1.

Column k=2 of A143328. - Alois P. Heinz, Jul 20 2013

Sequence in context: A191794 A191388 A194850 * A182024 A316474 A086661

Adjacent sequences: A062689 A062690 A062691 * A062693 A062694 A062695

Keyword

nonn,easy

Author

Gary L Mullen (mullen(AT)math.psu.edu), Jul 04 2001

Extensions

More terms from Sascha Kurz, Mar 25 2002

Status

approved