The Buses and the Mathematics

The image above shows an inside view of a bus, with a total number of c chairs. Consider that its chairs are numbered in ascending order and that this bus has an infinite space. A row of chairs is a line that contains chairs with numbers x, (x+1), (x+2), ..., y, where y is the number of the last chair of a row and x, the first. The numbers x and y also represent the chairs that are on the windows of that row. The total number of chairs is $c = 40$ .

Let $A_{k}n$ be the set that contains the sum of the numbers of each row, where k is the number of rows and n is the number of chairs on each row . Hence, the elements of $A_{k}n$ are { $S_{r1}$ , $S_{r2}$ , $S_{r3}$ ,..., $S_{rk}$ }, where $S_{r}$ is the sum of the numbers of a row r and rk is the last row of chairs. It is noticeable that k is directly dependent on the value of n. For instance, for n = 20, k must be 2, because the total number of chairs is 40.

This set has a property: sometimes, on a certain row, x and/or y are prime numbers. Yet, these rows that contain prime numbers on the windows sometimes also have $S_{r}$ = p , where p is also a prime number.

For 1 < n ≤ 10, what is the sum of all the possible values of p?

Details and Assumptions

• All the numbers are positive integers.

• For a certain value of n, all the rows must have a number of n chairs.

• At a row with prime numbers on the windows, x or y can be prime numbers. It is not a rule that it must be both of them.

• The elements of the set $A_{k}n$ depend on the value of n, and so does the value of k. Considering c = 30, a set $A_{2}15$ does not have the same elements as a set $A_{3}10$ .