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sums with number of divisors

Source: 7th QEDMO problem 5 (14. - 17. 1. 2010) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 9, 2021
number theoryDivisors

Problem Statement

For a natural number nn, let D(n)D (n) be the set of (positive integers) divisors of nn. Furthermore let d(n)d (n) be the number of divisors of n,n, that is, the cardinality of D(n)D (n). For each such nn, prove the equality ∑k∈D(n)d(k)3=(∑k∈D(n)d(k))2.\sum_{k\in D(n)} d(k)^3=\left( \sum_{k\in D(n)} d(k)\right) ^2.
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