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.

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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