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In the answers of another question (Effective Upper Bound for the Number of Prime Divisors) I found the following proposition:

The number of prime divisors counted with multiplicity is maximized for powers of $2$ and so

$$\Omega(n)\le\frac{\log n}{\log 2}=\log_2 n$$

Can someone give me a formal proof for this?

Paul Ostmann
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  • Well, that's obvious. For any number $n$ with any prime divisors, changing them all to 2 will result in a smaller $n$ with the same $\Omega$. – Ivan Neretin May 10 '18 at 15:31
  • Thank you. I‘m not a native speaker and thought powers of 2 meant raising all the prime devisors to the power of 2. – Paul Ostmann May 11 '18 at 06:42

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