Maximizing product of list of integers whose sum equals n Given a positive integer n, find the list of positive integers whose product is the largest among all the list whose sum is n. For example, if n is 4 the desired list is 2, 2 because 2x2 = 4 is larger than 1x1x1x1 = 1, 2x1x1 = 2 and 3x1 = 3, if n is 5, the desired list is 2, 3.
What is the desired list if n = 2001?
Actually, what is an algorithm to generate this list? After tried lots of examples, I discern (hope am right) for n &gt; 5 it's a list of 3's, and a remainder n % 3.
So, for n = 2001, I got a list of 667 3's.
Question
If Correct, what property of 3 makes it the ideal integer in that it composes (mostly) this list?.
Intuitively the function $(n/x)^x$ is maximized at $x=\frac{n}{e}$ and 3 is closest to $e$.