## 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.

### The solution to this problem requires strong mathematical programming.

This problem can be solved by first creating functions to check for practical numbers and sexy prime pairs. Then create a function to check for Engineer’s Paradises, then finally sum the first four paradises.

Before moving forward, let’s understand what Practical numbers and Sexy prime pairs are.

Practical Numbers:

According to Wolfram Mathworld, a practical number, also known as a panarithmic number or Egyptian number, is a positive integer n in which every smaller positive integer can be represented as a sum of distinct submultiples.

Sexy Prime pairs:

Two prime numbers that differ by 6 are known as sexy primes, since “Sex” is the Latin word for six.

Evaluating those for the calculation of an engineers’ paradise:

We can start by doing a calculation on lower numbers and iterating them with a computer until we find suitable engineers’ paradise. Then we loop through potential solutions and find the first four paradises using Python or a similar programming language.

Python is a good choice for this kind of problem because it has large-number handling built in, and also has several libraries that are useful for solving problems like these.

Once we have identified these 4 paradises, it is just arithmetic to sum them. The building of this code and forming up to the mathematical problem is a tedious task and requires a good handling of mathematical programming.

Please let me know if you would like further clarifications.

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