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More formally, find an asymptotic for $N\to\infty$ of $\frac{\sum_{1\le k\le N} M(k)}{N}$ where $M(p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}) = d_1+d_2+\cdots+d_k$

For example, $M(24) = M(2^3\cdot3) = 4$.

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Check out equation (3) here. Unsurprisingly, the first term in the series is the same as the one for the distinct divisor function $\omega(n)$, namely $\log \log n$.

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    Well, I supposed that the answer should be $\log \log N + constant$, thanks for a reference2011-07-12
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equation (20) http://mathworld.wolfram.com/PrimeFactor.html gives the exact answer as $\log \log N +B$ for some constant B. The link gives reference to what the value of B is.