What is the name for the function that gives the ratio of prime divisors to composite divisors of a given integer? Example, 5 gives (1,5:-) 2/0 = nan, 6 is (1,3:2,6) 2/2=1, 8 is (1:2,4,8) 1/3. Has anyone ever analysed this sort of thing before? Is there somewhete I can read up on it?
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Why are you treating $1$ as a prime? As $2$ as not a prime? – Erick Wong Sep 20 '15 at 05:14
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And of course there would be the question of 1 not counting as a prime. In my example perhaps I would have been better to call it composite divisors and other? And to pay attention to where I put that two? Its one in the morning here, that was human error – Ian Sep 20 '15 at 05:15
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I do not know of a name. Prime to overall is nicer, and average behaviour might be interesting. – André Nicolas Sep 20 '15 at 05:18
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http://math.stackexchange.com/questions/1217411/number-of-distinct-prime-divisors-of-an-integer-n-is-o-log-n-log-log-n I will have a look at this quesrion to see if therebisnsome insight – Ian Sep 20 '15 at 13:31
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https://en.m.wikipedia.org/wiki/Hardy–Ramanujan_theorem. Bingo, I do believe this is the answer I was looking for!
Ross Millikan
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Ian
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After some delay you can accept this answer so it doesn't show up unanswered. I fixed the link. If you click the link icon when posting the answer the site will put square brackets around the whole link and not get confused by the hyphen. – Ross Millikan Sep 20 '15 at 13:45
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$\omega(n)$ has a normal order, but the divisor function ($d(n)$ or $\tau(n)$) doesn't have a normal order for the following reason. By the Erdős-Kac theorem, there is a positive density of integers with $\omega(n) > \log\log n + \sqrt{\log \log n}$, and $\tau(n) \ge 2^{\omega(n)}$. Conversely there is also a positive density of integers with $\Omega(n) < \log\log n - \sqrt{\log \log n}$, and $\tau(n) \le 2^{\Omega(n)}$. So $\tau(n)$ often deviates significantly far on each side of $2^{\log\log n}$. – Erick Wong Sep 20 '15 at 23:11