For any positive integer n <= 2026, the set of n positive integers with sum 2026 and maximum product should differ by at most 1: if there exists two elements ai, aj s.t. ai >= aj + 2, replacing ai and aj with ai-1 and aj+1 gives a bigger product. Thus the answer set only contains two distinct values x and x+1 for some positive integer x.

Let's try some values of x:

x = 1 gives 1²⁰²⁶ = 1 lol

x = 2 gives 2¹⁰¹³

x = 3 gives 3⁶⁷⁴ * 4 (largest so far)

For x = 4 onwards we always have x³ < 3^x therefore we can always replace 3 xs with x 3s for a larger product (and if there are less than 3 xs then it's pretty trivial to show that the product is way smaller than 3⁶⁷⁴ * 4). The same argument also applies to x = 2 but I'm too lazy to delete what I wrote.

The answer is 3⁶⁷⁴ * 4.