If we change the condition to $a_t\leq n$ it can be shown that\[A(n)=n-\frac{n}{\log n}+o\left(\frac{n}{\log n}\right)\]via a greedy decomposition (use $n$ as often as possible, then $n-1$, and so on). Other questions can be asked for other restrictions on the sizes of the $a_t$.

Cambie has observed that a positive answer follows from the result above with $a_t\leq n$, simply by pairing variables together, e.g. taking $a_i'=a_{2i-1}a_{2i}$ (and the lower bound follows from Stirling's approximation).

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