Let's call two integers
approximately equal if they differ by at most 1.
How many different ways are there to write 2017 as a sum of two or more
positive integers which are all
approximately equal to each other?
The order
of terms does not matter: two ways which only differ in the order of terms are counted as one.
Assume a sum consists of N summands. Then for each integer N there is exactly one sum. Let q=floor(2017/N) and r=2017 mod N. q and q+1 are the approximately equal integers. Then the sum consisting of N summands consists of q occurring N-r times and q+1 occurring r times.
There are 2016 positive integers greater than 1 and not exceeding 2017, therefore there are 2016 different ways to write 2017 as a sum of two or more positive integers which are all approximately equal.
Solution
