RE: [math-fun] First time harmonic series exceeds 100

The harmonic number H(n) := 1 + 1/2 + 1/3 + . . . + 1/n has the interesting property that although it -> oo as n does, it's never an integer. So let H(n) = p(n)/q(n) in lowest terms. (I have full confidence that p and q are already in the OEIS.) What is known about the factorizations of the p(n)'s and q(n)'s ? In particular, are their factorizations known or believed to have statistical properties atypical of numbers of the same size (re the number, size, and exponents of their prime factors) ? Do the factors of p(n) (or q(n)) have special number-theoretical properties? (I.e., what is known about the primes -- if any -- that *never* occur as a factor of any p(n) (q(n)) ? --Dan