In a recent exercise, we discussed the problem of computing the nearly square divisors of a composite number n = a · b, with a ≥ b, such that the difference ab is as small as possible. We gave two solutions: The first solution enumerated the divisors one-by-one, by trial division counting from 1, until reaching the square root, which is fast if n is small but slow in general. The second solution factored n, computed the divisors, the picked the two divisors in the middle of the list, which is fast in general but slow if n is highly composite and thus has a large number of divisors.

In a comment on that exercise, Matthew Arcus, who is a regular reader and commentor at Programming Praxis, gave a splendid answer to the problem; it relies on factoring n, but is fast even if n has a large number of divisors. His algorithm reduces the multiplication of the factors to addition of their logarithms, which means that the knapsack algorithm can be used to find the greatest sum less than the logarithm of the square root.

Your task is to implement Matthew’s algorithm to find the nearly-square divisors of a number. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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