The Lagarias arithmetic derivative D(n) is defined for integers such that the derivative of a prime number is 1: D(p) = 1. It also satisfies the Leibniz formula, which resembles the product rule of calculus:
D(mn) = mD(n)+nD(m)
For example, the arithmetic derivative of 15 is 3D(5) + 5D(3) = 3x1 + 5x1 = 8. The Leibniz formula also implies that D(1) = 0.
Write a function to compute the arithmetic derivative of a natural number.
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