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→Formulas at prime powers: Explain further an imporant result |
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:<math>\sigma_0(n)=\prod_{i=1}^r (a_i+1).</math>
This result can be directly deduced from the fact that all divisors of <math>n</math> are uniquely determined by the distinct tuples <math>(x_1, x_2, ..., x_i, ..., x_r)</math> of integers with <math>0 \le x_i \le a_i</math> (i.e. <math>a_i+1</math> independent choices for each <math>x_i</math>).
For example, if ''n'' is 24, there are two prime factors (''p<sub>1</sub>'' is 2; ''p<sub>2</sub>'' is 3); noting that 24 is the product of 2<sup>3</sup>×3<sup>1</sup>, ''a''<sub>1</sub> is 3 and ''a''<sub>2</sub> is 1. Thus we can calculate <math>\sigma_0(24)</math> as so:
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