A395219 - OEIS

%I #19 Apr 30 2026 00:24:42

%S 6,7,8,5,0,2,2,2,1,8,6,6,3,2,3,1,4,2,0,6,4,4,3,8,8,9,5,4,3,8,9,8,7,5,

%T 6,8,9,0,1,7,5,1,6,5,9,7,4,1,3,1,7,2,9,2,3,6,0,3,6,5,4,2,9,4,2,5,1,4,

%U 5,4,0,3,8,8,8,0,6,1,4,0,3,4,9,9,8,7,6,0,7,7,1,3,0,9,9,4,9,9,8,3,0,2,3,5,7

%N Entropy of the zeta probability distribution with parameter s=3.

%C Sum_{k>=1} (-m(k)*log(m(k))), where m(k) is the mass distribution function for zeta distribution, m(k) = 1/k^s/zeta(s), with s = 3.

%H Stanislav Sykora, <a href="/A395219/b395219.txt">Table of n, a(n) for n = 0..999</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Zeta_distribution">Zeta distribution</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Entropy_(information_theory)">Entropy (information theory)</a>.

%F Equals -Sum_{k>=1} (log(1/k^3/zeta(3))/k^3/zeta(3)).

%F Equals log(zeta(3))-3*zeta'(3)/zeta(3).

%e 0.678502221866323142064...

%o (PARI) \p200

%o s=3; zs=zeta(s);

%o a=log(zs)-(s/zs)*zeta'(s);

%o precision(a, 105)

%Y Cf. A395218 (s=2), A395220 (s=4).

%Y Cf. A013661, A073002.

%K nonn,cons

%O 0,1

%A _Stanislav Sykora_, Apr 18 2026