We examine a class of interesting functions used in number theory.

THE TAU FUNCTION

The function τ(n) counts how many divisors n has. This count includes 1 and n. (τ is a Greek letter and is called “tau.”) The first few values are τ(1) = 1, τ(2) = 2, τ(3) = 2, τ(4) = 3, τ(5) = 2, τ(6) = 4, τ(7) = 2, τ(8) = 4, τ(9) = 3, and τ(10) = 4.

Let n have the prime decomposition
. It is easy to see that m is a divisor of n if and only if
where 0 ≤ c_{1} ≤ e_{1}, 0 ≤ c_{2} ≤ e_{2}, 0 ≤ c_{3} ≤ e_{3}, …, 0 ≤ c_{r} ≤ e_{r}. (This can be stated more tersely as “0 ≤ c_{i} ≤ e_{i} for i = 1, 2, …, r.”) As an example, the divisors of 2^{3}3^{2}7^{1} = 8 × 9 × 7 = 504 have the form 2^{a}3^{b}7^{c}, where a = 0, 1, 2, or 3; b = 0, 1, or 2; and c = 0 or 1. In general, there are e_{1} + 1 values ...

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