Positively Described Numbers - Part 2

Surely you've heard of the perfect numbers, numbers that have the property that the sum of the proper factors of the number are equal to the number. The first few examples of perfect numbers are \(6,\) \(28,\) and \(496.\) Because the sum of the proper factors of perfect numbers is equal to the number, the sum of \(\textit{all}\) of the factors of the number is equal to twice the number. Because of this, perfect numbers are called \(2\text{-perfect.}\)

A number is \(k\text{-perfect}\) if the sum of the factors of the number is equal to \(k\) times the number. Let \(\{a_1,a_2,a_3\ldots,a_n,\ldots\}\) be the set of numbers that are the smallest \(k\text{-perfect}\) number for increasing values of \(k,\) starting at \(1.\) \(a_n\) is the smallest \(n\text{-perfect}\) number. What is the sum of the digits of this sum? \[\sum_{i=1}^5a_i\] \[\text{...................................................................................................................}\] \(\textbf{Hint:}\) It will take a really long time for you to solve this directly. How can you optimize the amount of numbers you need to check?

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