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# 6NUMBER THEORETIC FUNCTIONS

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 ≤ cr  ≤ er . (This can be stated more tersely as “0 ≤ ci  ≤ ei for i = 1, 2, …, r.”) As an example, the divisors of 233271 = 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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