As a next application of arithmetic functions we provide a general formula for the number of positive divisors \(\tau(n)\).

Proposition: Calculating the Number of Positive Divisors

Let \(n > 1 \) be a natural number with the factorization. \[n=\prod_{i=1}^\infty p_i^{e_i}.\]

Then, the number of positive divisors \(\tau(a)\) can be calculated as

\[\tau(a)=\prod_{i=1}^\infty (e_i+1).\]

Example:

\(n=877800\) has \(192\) positive divisors, since

$$n=877800=2^3\cdot 3^1\cdot 5^2\cdot 7^1\cdot 11^1\cdot 13^0\cdot 17^0\cdot 19^1\cdot\prod_{i=9}^\infty p_i^0$$ and \[\tau(877800)=4\cdot 2\cdot 3\cdot 2\cdot 2\cdot 1\cdot 1\cdot 2\cdot\prod_{i=9}^\infty 1=192.\]

Proofs: 1

Proofs: 1


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927