Let me illustrate. If I pick 12 the factors are {1, 2, 3, 4, 6 and 12}. The number of factors of those numbers are {1, 2, 2, 3, 4, 6}. The magic part... 1

^{3}+2

^{3}+2

^{3}+3

^{3}+4

^{3}+6

^{3}=324... and (1+2+2+3+4+6)

^{2}= (18)

^{2}= 324...

This is related to the well known relation that for any string of consecutive integers {1,2,3, ...n} the sum of the cubes is equal to the square of the sum. (student's should prove this by induction).

I came across this recently at a blog called Alasdair's Musing. He gives credit to Joseph Liouville. He has a nice proof of the relation using Cartesian cross products of sets. Yes, children, you want to know what that means.

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