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This is my first question.

Let $a_1, a_2,\ldots, a_k$ be natural numbers $\leq n$ with $m$ prime factors.

Let $p_1, p_2, \ldots, p_r$ be the prime numbers $\leq n$.

Let $C_{m,n} = \frac{\sum_{i=1}^{k}a_i}{\sum_{i=1}^{r}p_i}$ Does the following limit exist when $m\geq 2$? Obviously, when $m=1$, $C_{1,n}=1$.

$\lim_{n\to \infty} C_{m,n}$

I have some guesses but not defendable enough.

My best,

The FM.

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    @GerryMyerson, thanks a lot. In addition to trying to have a paper and pen proof attempt, I had better try writing some programs and make better guesses as well.2012-06-04

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For $m\gt1$, there are more numbers with $m$ prime factors than there are primes, in the following sense:

Hardy and Wright, Theorem 437, prove that the number of integers up to $x$ with exactly $m$ prime factors (and it doesn't matter whether the primes are distinct or not) is asymptotic to ${x(\log\log x)^{m-1}\over(m-1)!\log x}$ They attribute the result to Landau, 1900. Now the number of primes is asymptotic to $x/(\log x)$, so the ratio is $(\log\log x)^{m-1}/(m-1)!$, which of course goes to infinity with $x$.

Meanwhile, the sum of the primes up to $x$ is asymptotically $x^2/(2\log x)$ although I haven't found a wholly trustworthy cite for this. I'm sure (although I'm not up to proving it) that the sum of the almost-primes will be bigger by that same factor of $(\log\log x)^{m-1}/(m-1)!$

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    AT Gerry Myerson, thank yo$u$ for the clear reference and the discussion of $y$o$u$r ideas. I will also try to check your conjecture in the last paragraph.2012-06-05