# brilliant.org

 Random Link ¯\_(ツ)_/¯ Feb 6, 2021 » 021. Amicable Numbers 8 min; updated Feb 6, 2021 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 $$d(220) = 284$$.... Jun 20, 2020 » 01. Sum of Powers 2 min; updated Jun 20, 2020 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: $$S_n = n + (n-1) + (n-2) + … + 1$$...