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Number of the divisors: Difference between revisions

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The '''number of divisors''' of a number (<tt>''d(n)''</tt>) can be found from its prime factorization as the product of the by-one incremented exponents of all present prime bases.
{{Wikipedia| Table of divisors }}


* If the number <math>n</math> has the [[prime factorization]]
The '''number of divisors''' {{w|Divisor function|d(''n'')}} of a number ''n'' can be found from its [[prime factorization]] as the product of the by-one incremented exponents of all present prime bases.
:: <math>n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r},</math>
: then:<ref>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&nbsp;273, p.&nbsp;239.</ref>
:: <math>d(n) = (e_1+1)(e_2+1) \dotsm (e_r+1)</math>
* for [[coprime]] numbers <math>m</math> and <math>n</math> applies:
:: <math>d(mn) = d(m)\cdot d(n)</math>


== Some Examples ==
If the number ''n'' has the prime factorization
 
<math>\displaystyle n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r},</math>
 
then:<ref>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&nbsp;273, p.&nbsp;239.</ref>
 
<math>\displaystyle d(n) = (e_1 + 1)(e_2 + 1) \dotsm (e_r + 1)</math>
 
For {{w|Coprime integers|coprime}} numbers ''m'' and ''n'' it follows that
 
<math>\displaystyle d(mn) = d(m) \cdot d(n)</math>
 
== Examples ==


{| class="wikitable"
{| class="wikitable"
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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>
|}
|}
== See also ==
* [[Highly composite equal division]]


== References ==
== References ==
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== External links ==
== External links ==


* [https://en.wikipedia.org/wiki/Table_of_divisors Table of divisors &#45; Wikipedia]
* [https://en.wikipedia.org/wiki/Divisor_function Divisor function &#45; Wikipedia]
* {{OEIS|A000005}}
* {{OEIS|A000005}}


[[Category:Math]]
[[Category:Math]]
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