Piles of Plates
Problem 688
We stack plates into non-empty piles where each pile is a different size. Define to be the maximum number of plates possible in the smallest pile. For example when and the piles is the best that can be done and so . It is impossible to divide 10 into 5 non-empty differently-sized piles and hence .
Define to be the sum of for all possible pile sizes . For example .
Further define . You are given .
Find giving your answer modulo .