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
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.
Crossrefs
Programs
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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);
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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 *) -
PARI
a(n)=sum(m=1,n, 1/m* sumdiv(m, d, moebius(d)*2^(m/d) ) ); /* Joerg Arndt, Jul 04 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
Extensions
More terms from Sascha Kurz, Mar 25 2002
Comments