I have found a function to create the lowest common multiples for the first $n$ positive integers:
$$\text{lcm}(n):=\prod\_{m=2}^{n}C\_m(1),$$
where $C\_m(1)$ is the [cyclotomic polynomial](http://mathworld.wolfram.com/CyclotomicPolynomial.html) of order $m$ and $n\geq2.$ 


The *Mathematica* functions:



```
lcm[n_] := Product[Cyclotomic[m, 1], {m, 2, n}]
\[Psi][x_] := Sum[Log[Cyclotomic[n, 1]], {n, 2, x}]
Table[Product[Cyclotomic[n, 1], {n, 2, m}], {m, 0, 28}],  

```

the last of which produces this sequence at [OEIS A003418](http://oeis.org/A003418)


Is this a known function?



New identity for lcm of the first n integers and the second Chebyshev function
Since $\log C\_m(1) = \Lambda(m)$ for $m \ge 2$, where $\Lambda$ is the Von Mangoldt function, this seems to be a restatement of the identity 


$$\mathrm{lcm} \{1,\ldots,n\} = \exp \psi(n) = \exp\bigl( \sum\_{m=1}^n \Lambda(m) \bigr) = \exp\bigl( \sum\_{m=2}^n \log C\_m(1) \bigr).$$


