Problem 263

An engineers’ dream come true

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.

工程师美梦成真

数6的约数为：1、2、3和6。
从1到6的每个数都能写成6的不同约数的和：
1=1, 2=2, 3=1+2, 4=1+3, 5=2+3, 6=6.
如果从1到n的每个数都能写成n的不同约数的和，n就被称为实用数。

如果一堆连续素数的差为6，则被称为性感数对（因为”sex”（性）恰好是”six”（六）的拉丁语写法）。第一个性感数对是(23, 29)。

我们偶尔可能会发现性感数对三元组，也就是三组连续的性感数对，其中每一组的第二个数恰好是后一组的第一个数。

如果数n满足：

(n-9, n-3)、(n-3,n+3)、(n+3, n+9)构成一个性感数对三元组，而且
数n-8、n-4、n、n+4和n+8都是实用数，

我们就称之为工程师的天堂数。

找出前四个工程师的天堂数，并求它们的和。