Consider all possible values for $(a_1,\ldots,a_n)$ (where each $a_i$ may be a positive integer or 0) when $a_1 + \cdots + a_n = n$.
Consider $(a_j,\ldots,a_k)$ the values in $(a_1,\ldots,a_n)$ which are non-zero.
Given a set of values for all $a_i$, $j\le i\le k$ can we check whether they maximise $\operatorname{lcm}(a_j,\ldots,a_k)$ without testing all other cases?
Is there a computationally viable way to always pick values for all $a_i$ which maximise $\operatorname{lcm}(a_1,\ldots,a_k)$?
The sequence is given in OEIS A000793. It is called Landau's function and starts $$1, 1, 2, 3, 4, 6, 6, 12, 15, 20, 30, 30, 60, 60, 84, 105, 140, 210, 210,\\ 420, 420, 420, 420, 840, 840, 1260, 1260, 1540, 2310, 2520, 4620, 4620,\\ 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 27720, 30030,\\ 32760, 60060, 60060, 60060, 60060, 120120$$ It grows about as $e^{\sqrt{n \log(n)}}$. A number of references are given.