  Let $Q(n)$ denote the number of integers $1 \leq q \leq n$ whose prime
factorization $q= \prod^{t}_{i=1}p^{a_i}_i$ satisfies $a_1\geq a_2\geq \ldots
\geq a_t$. Hardy and Ramanujan proved that $$ \log Q(n) \sim
\frac{2\pi}{\sqrt{3}} \sqrt{\frac{\log(n)}{\log\log(n)}}\;. $$ Before proving
the above precise asymptotic formula, they studied in great detail what can be
obtained concerning $Q(n)$ using purely elementary methods, and were only able
to obtain much cruder lower and upper bounds using such methods.
  In this paper we show that it is in fact possible to obtain a purely
elementary (and much shorter) proof of the Hardy--Ramanujan Theorem. Towards
this goal, we first give a simple combinatorial argument, showing that $Q(n)$
satisfies a (pseudo) recurrence relation. This enables us to replace almost all
the hard analytic part of the original proof with a short inductive argument.
