Ortoedres amb longitud d'arestes enteres / Cuboids with integer length edges

Abstract

In this article we study the number of different cuboids $\mathcal{O}(N)$ that can be built with an arbitrary number $N$ of equal cubes. This problem is equivalent to find the number of different cuboids of volume $N$ with integer length edges. We obtain an iterative method to calculate the value of $\mathcal{O}(N)$ for any $N$. Using this method we obtain an explicit formula when $N$ is the product of two powers of prime numbers. The bidimensional case is also studied and we give a general formula to determine the number of different rectangles that can be built with an arbitrary number of equal squares.