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x equals any whole number.

y equals the number of prime factors of x.

You plot those points, then find a line of best fit.

What would the equation for that line be?

Also; why?

$x = 48$

$y = 5$

Because $48 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$

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    If we do insist on a line, $y=0$ seems like the best candidate :)2011-09-07

1 Answers 1

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It's easiest to think in terms of the partial sums of this function (number of prime factors of 1, plus number of prime factors of 2, plus ..., plus number of prime factors of n). Call this function $f(n)$. What does the best curve for $f(n)$ look like? Well, $f(n) \approx n/2 + n/3 + n/4 + n/5 + n/7 + n/8 + n/9 + \cdots + n/p^k$ for $p$ prime, $p^k < n$. Why? Because approximately $n/2$ of the numbers less than $n$ are divisible by 2 once, $n/3$ are divisible by 3 once, $n/4$ are divisible by 2 twice, etc.

So what does that sum look like? Well, $1/p + 1/p^2 +\cdots + 1/p^k = \frac{p^k - 1}{p^k (p-1)}$, which is approximately $1/(p-1)$. So

$f(n) \approx \sum_{p\leq n} \frac{1}{p-1}$

where the sum is over primes.

Your function is approximately the derivative of $f(n)$, which can be approximated by $1/(n-1)$ times the probability that $n$ is prime, or roughly $1/n \ln n$.

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    Ok. Nice. I'll vote it up.2011-09-07