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An engineers' dream come true

Problem 263

Published on Saturday, 7th November 2009, 01:00 am; Solved by 642

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.