In the vibrant land of Numeria, where numbers are not just symbols but the essence of life itself, the annual Festival of Sums is a spectacle of mathematical prowess. This year, the festival introduces a new challenge, captivating the minds of Numerians: to craft summations that not only meet but can also exceed a specified sum $$$S$$$, using the orderly sequence of multiples of a chosen number $$$P$$$. This challenge embodies Numeria's celebration of abundance and precision, inviting participants to demonstrate their skill in navigating the realms of numbers beyond the exactness, into the realm of surplus.

As a contender in this celebrated festival, you are presented with an array of numbers from $$$1$$$ to $$$N$$$. Your quest involves solving $$$Q$$$ puzzles, for each puzzle you must find how many unique ways you can select two distinct positive integers $$$i$$$, and $$$j$$$ such that $$$j - i \leq 10^3$$$, and sum all numbers $$$kP$$$ where $$$i \leq k \leq j$$$ and $$$kP \leq N$$$ for a given prime number $$$P$$$ such that the total sum is at least $$$S$$$. Two selections are different if any of the values for $$$i$$$ and $$$j$$$ differ.