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Let $\omega(n) = \sum_{p \mid n} 1$. Robin proves for $n > 2$, \begin{align} \omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}}. \end{align} Is there a similar tight effective upper bound for $\Omega(n) = \sum_{p \mid n} \text{ord}_{p}(n)$ or at least an upper bound in terms of $\omega(n)$?

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    The original paper uses the constant 1.45743, and the bound applies for $n \geq 3$, with equality for $N_{47}$, which I believe to be the $47$th prime.2014-11-25

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

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

and since it is exactly equal for infinitely many $n$ it is also the tighest possible bound.