Sets, that have $\operatorname{LCM}\left(|c_1|,\dots,|c_p|\right)=\sum_{k=1}^p |c_k|$

Question

I found that the least common multiple of the sizes of conjugacy classes $c_k$ of the symmetric group $S_n$ is equivalent to $n!$ the order of the group. Equivalently the sum of all $c_k$ is also $n!$. I checked $n<7$ and $n=8$.

So we have $\operatorname{LCM}\biggr(|c_1|,\dots,|c_p|\biggr)=\sum_{k=1}^p |c_k|$. Do these sets of integers $\{|c_k|\}$ have a special name and further properties, occurences or application?

EDIT

The sets of divisors of perfect numbers would fit, since e.g. $$\operatorname{LCM}\biggr(1,2,7,14\biggr)=28=1+2+4+7+14,$$ therefore a related question is this one.