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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 $f(n)$.

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    Generally sequences with the same asymptotic growth can have wildly different prime factor frequencies. If you added the condition that it is a (perhaps strong) [divisibility sequence](http://en.wikipedia.org/wiki/Divisibility_sequence), then I believe there would be more interesting answers...2011-10-06

2 Answers 2

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The paper The least common multiple of a quadratic sequence by Javier Cilleruelo, Compositio Mathematica (2011), 147: 1129-1150 gives the answer when $f$ is a quadratic polynomial. From the abstract:

For any irreducible quadratic polynomial $f(x)$ in $\mathbb{Z}[x]$ we obtain the estimate $\log\mathrm{LCM} (f(1),...,f(n)) = n\log n + B\,n + o(n)$ where $B$ is a constant depending on $f$.

This is a link to the paper in Arxix.

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Konowing asymptotics of $f$ is not enough. Consider functions $f_1(n)=2^n$ and $f(n)=p_1p_2\ldots p_n\;$, where $p_n$ is the $n$-th prime number.

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    Consider $f_1(n) = 2^n$ and $f_2(n) = 2^n-1$, then. Then the LCM of the second sequence grows much faster than that of the first.2011-10-04