The algorithm to be used is:

- Sort the set into assenting order
- $x_1 = s_1$
- $x_i = gdc(x_{i-1},s_i)$
- $LCM = \frac{\Pi_{i\in S}i}{x_n}$

What I'm looking for is expected run time as a function of $\sum_{i\in S}i$

As a starting point $|S| \leq \sum_{i\in S}$ and GDC is $O(ln(n)^2)$ so an upper bound should be $O(n\ln(n)^2)$.