A well known theorem of Gauss says that any natural number $n$ may be written as the sum of three triangular numbers - $$ n={a_{1} \choose 2}+{a_{2} \choose 2}+{a_{3} \choose 2} $$
The following question came up over the course of my work - Does there exist a (slowly growing) function $\omega(n)$ which tends to infinity with $n$ such that any natural number $n$ may be written in the following form - $$ n=\sum_{i=1}^{k}{a_{i} \choose 2}\mbox{ subject to }\prod_{i=1}^{k}a_{i}\leq\frac{n}{\omega(n)} $$
If we choose the largest possible $a_{1}$, then the largest possible $a_{2}$ etc; we can guarantee $\prod_{i=1}^{k}a_{i}\leq Cn$ for some absolute constant $C$ (NOT QUITE - SEE BELOW). The question then is, can we do any better?
[EDIT 1: In my rush to post this, I did not check my calculations very well - if you actually do what is described, you only get $\prod_{i=1}^{k}a_{i} \leq 2^{\log_2\log_2 n}n^{1/2+1/4...} = O(n\log n)$ (as Erick Wong and Emil Jeřábek pointed out). The question of how small you can make $\prod_{i=1}^{k}a_i$ still remains.]
[EDIT 2: Will Jagy has computed what the best possible product for $n\leq10^7$ and based off this data, the existence of an $\omega$ as asked above seems unlikely. That for some infinite family of natural numbers $n$, the product must be $\Theta(n)$ is a plausible conjecture - though how one might prove something like that is not clear to me.]
