one of the most prominent functions of the first $n$ natural numbers is the factorial $n!$ that denotes their product.


Today however I wondered whether the least common multiple $\mathrm{lcm}(n):=\mathrm{lcm}(\{i\in\mathbb{N}\,|\,1\leqq i\leqq n\})$ of the first $n$ natural numbers has already been the subject of mathematical studies.  

The first elements of their sequence are $1,\, 2,\, 6,\,12,\,60,\,60,\,420,\,840,\,2520,\,2520,\,27720,\, \dots$ and it is A003418 – in the **OEIS**, as Christian Bernert pointed out in his comment.



> 
> **Questions:**
> 
> 
> * is there already an established name for $\mathrm{lcm}(n)$?
> * is there an agreed upon notation?
> * does it have an extension for non-integral complex values, analogous to the gamma function?
> * what can be said about bounds on, resp. growth-rate of the number of consecutive equal values apart from their relation to prime-gaps?
> 
> 
> 


It is not hard to see that $\mathrm{lcm}(n)$ equals the *procuct of maximal prime powers less or equal n*



Name and properties of $\mathrm{lcm}(\{1,\,\cdots,\,n\})$
The quantity $\operatorname{lcm}(n)$ is equal to $e^{\psi (n)}$, where $\psi$ is the second [Chebyshev function](https://en.wikipedia.org/wiki/Chebyshev_function). This function is well studied, and the prime number theorem is equivalent (indeed, usually proved using this equivalence) to the fact that $\psi (x)\sim x$.


