## An engineers' dream come true

### Problem 263

Published on Saturday, 7th November 2009, 01:00 am; Solved by 697; Difficulty rating: 75%
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,

Find the sum of the first four engineersâ€™ paradises.