Well as nobody else has answered this I will tell you what I know though its not a complete solution.

The arithmetic-geomentric mean inequality tells us that if there are

numbers involved, and that these are

, then:

Regarding the lefthand side of this inequality as a function of N this has a maximum

when

. So we have that the maximum posible value of the product of an integer partition of

is less than or equal

.

The partition

has product

and I would not be supprised if this were the maximum, but have not done enough to convince even myself that this is the case.

RonL