Let $n\ge 1$ and $G$ a group of order $n$. Determine the order of every element of $G$ and list the orders increasing.
Can the number of possible order-sequences be determined efficiently ?
For example, a group of order $6$ has the possible order-sequences
$$1,2,2,2,3,3$$ and $$1,2,3,3,6,6$$
so there are two possible order-sequences for $n=6$.
The number of order-sequences gives a lower bound for the number of groups of order $n$.
Additional question : Can the number of groups with a given order-sequence be determined efficiently ?