The Number Theoretic Omega Function and Summations Involving the Exponents of Prime Numbers in the Factorization of Factorials

‎The aim of this paper is to study the balancing of prime factors and their exponents in the standard factorization of $n!$ into primes‎. ‎We obtain explicit approximation for the sums $\sum_{p\leqslant x(n)}\u_p(n!)$ and $\sum_{p\leqslant x(n)}\u_p(n!)\log p$ for each boundary function $x(n)$ with $2\leqslant x(n)\leqslant n$‎. ‎Also‎, ‎we estimate sums involving the exponents of prime factors in the factorization of factorials and conclude that the sum of exponents of primes not exceeding $\e^{\sqrt{\log n}}$ is asymptotic to the sum of the exponents of other primes in the factorization of $n!$ into primes‎, ‎as $n\rightarrow \infty$‎. ‎It is also shown that the product of primes not exceeding $\sqrt{n}$ with their multiplicity is asymptotic to the product of other primes in the factorization of $n!$ with their multiplicity‎, ‎in logarithmic scale‎.