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### In which orders can the numbers of prime factors of consecutive integers be?

Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given ...

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### Relation between the densities of two functions and their additive convolution?

Suppose that for two integral-valued arithmetic functions $f_i$ ($i=1,2$), the following values are known:
$$ \lim_{n\to \infty} \frac{ \{n:f_i(n) \text{ is odd}\}\cap \{0,1,\ldots, n-1\} }{n}.$$
(In ...