Problem Archive

Consider the number $6$. The divisors of $6$ are: $1,2,3$ and $6$.
Every number from $1$ up to and including $6$ can be written as a sum of distinct divisors of $6$:
$1=1$, $2=2$, $3=1+2$, $4=1+3$, $5=2+3$, $6=6$.
A number $n$ is called a practical number if every number from $1$ up to and including $n$ can be expressed as a sum of distinct divisors of $n$.

A pair of consecutive prime numbers with a difference of six is called a sexy pair (since "sex" is the Latin word for "six"). The first sexy pair is $(23, 29)$.

We may occasionally find a triple-pair, which means three consecutive sexy prime pairs, such that the second member of each pair is the first member of the next pair.

We shall call a number $n$ such that :

• $(n-9, n-3)$, $(n-3,n+3)$, $(n+3, n+9)$ form a triple-pair, and
• the numbers $n-8$, $n-4$, $n$, $n+4$ and $n+8$ are all practical,

an engineers’ paradise.

Find the sum of the first four engineers’ paradises.