Let us write $M(n)$ for $\operatorname{lcm}(1,\dotsc,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, here of the first $n$ positive integers, and its logarithm is a Chebyshev function.)

It also seems that $2^n \lt M(n) \lt 3^n$ for $n \gt 6$ and even $2^n \leq M(n+1) \leq 3^n$ for $n \geq 0$. Are these relations true, and are there combinatorial proofs of either?

Additionally, do these inequalities appear in the combinatorial literature?

Gerhard "Interest In LCM Is Growing" Paseman, 2020.05.19.

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