Asymptotics of LCM

Solution 1:

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.

Solution 2:

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.