# 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 $$A_{3}10$$.

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