Number of the divisors: Difference between revisions
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just a note about an interesting function |
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| <math>d(30) = (1+1)(1+1)(1+1) = 8</math> | | <math>d(30) = (1+1)(1+1)(1+1) = 8</math> | ||
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== See also == | |||
* [[Highly composite equal division]] | |||
== References == | == References == | ||
Revision as of 21:49, 22 November 2022
The number of divisors of a number (d(n)) can be found from its prime factorization as the product of the by-one incremented exponents of all present prime bases.
- If the number [math]\displaystyle{ n }[/math] has the prime factorization
- [math]\displaystyle{ n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r}, }[/math]
- then:[1]
- [math]\displaystyle{ d(n) = (e_1+1)(e_2+1) \dotsm (e_r+1) }[/math]
- for coprime numbers [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] applies:
- [math]\displaystyle{ d(mn) = d(m)\cdot d(n) }[/math]
Some Examples
| Number | Prime factorization | Number of divisors |
|---|---|---|
| 8 | [math]\displaystyle{ 8 = 2^3 }[/math] | [math]\displaystyle{ d(8) = (3+1) = 4 }[/math] |
| 12 | [math]\displaystyle{ 12 = 2^2 \cdot 3 }[/math] | [math]\displaystyle{ d(12) = (2+1)(1+1) = 6 }[/math] |
| 30 | [math]\displaystyle{ 30 = 2 \cdot 3 \cdot 5 }[/math] | [math]\displaystyle{ d(30) = (1+1)(1+1)(1+1) = 8 }[/math] |
See also
References
- ↑ G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. 4. Edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1, Theoreme 273, p. 239.