  Let omega(n) be the number of distinct prime factors dividing n and m > n
natural numbers. We calculate a formula showing which prime numbers in which
intervals divide a given binomial coefficient. From this formula we get an
identity omega(binom(nk)(mk))=sum_i (pi(k/b(i))- pi(k/a(i))) + O(sqrt(k)).
Erdoes mentioned that omega(binom(nk)(mk))= log n^n/(m^m (n-m)^(n-m)) k/log k +
o(k/log k).
  As an application of the above identities, we conclude some well-known facts
about the distribution of the primes and deduce for all natural numbers k an
expression (also well-known) log k = sum_i a_k(i) which generalizes log 2 =
sum_i^(infty) (-1)^(j+1) / j.
