I recently solved a problem which asked me to show that all numbers under $n!$ can be expressed as the sum of at most $n$ factors of $n!.$ The proof of this is a simple application of mathematical induction. My question is: for a given $n$, what is the smallest number which cannot be expressed as the sum of $n$ factors of $n!$? Any n (not necessarily distinct) factors are allowed to be used. Therefore for $n=3,$ $12=6+6=6+3+3$ are both valid.
Clearly, any number larger than $n*n!$ cannot be expressed as such a sum of factors, as the largest factor of $n!$ is $n!$ itself. Therefore, we are able to see that such a number does indeed exist.
I have also tried to do the problem for small cases. For example, if $n=2,$ then the smallest inexpressible value is $5,$ and if $n=3,$ then the smallest inexpressible value is $16.$ Finally, if $n=4,$ the smallest inexpressible value is $65.$ I cannot seem to be able to generalize these values, however.