Let $\operatorname{LCM}(x_1,x_2,\ldots,x_n)$ be the least common multiple of the integers $x_i$.
How can one find the asymptotics of $\operatorname{LCM}(f(1),f(2),\dots,f(n))$ as $n$ approaches infinity if one knows the asymptotics of the strictly increasing function on the integers $f(n)$?
Edit: Is there some result if one assumes $f(n)$ has natural density in the primes ie is prime with probability $1/\ln(f(n))$, has an average of $ln(f(n))$ factors, $ln(ln(f(n))$ prime factors, and a $6/\pi^2$ probability to be squarefree.
And how to prove these average properties ?
Edit2: Instead of above estimates use: For every strictly increasing $f(n)$, consider instead the question for $g(f(n))$, which is a uniformly random integer in the range $(0.99f(n),1.01f(n))$ say, alternatively $(f(n)-1000,f(n)+1000)$.
Then im looking for asymptotics of $LCM[g(f(1)),g(f(2))...g(f(x))]$ as $x\rightarrow\infty$, given $f(n)$.