brilliant.org

Problem Statement Let \(d(n)\) be defined as the sum of proper divisors of \(n\) (numbers less than \(n\) which divide evenly into \(n\)). If \(d(a) = b\) and \(d(b) = a\), where \(a \neq b\), then \(a\) and \(b\) are an amicable pair and each of \(a\) and \(b\) are called amicable numbers. For example, the proper divisors of \(220\) are \(1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110\); therefore \(d(220) = 284\)....

The sums of powers, \( \sum_{k=1}^{n} k^a \), can be computed more efficiently if we have a closed formula for them. Sum of powers for a = 1, 2, 3 $$ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} $$ One [intuitive] derivation is presented at brilliant.org : $$ S_n = 1 + 2 + 3 + … + n $$ … can be reordered as: $$ S_n = n + (n-1) + (n-2) + … + 1 $$...