Template:Number of divisors

The following documentation is located at Template:Number of divisors/doc.

The {{number of divisors}} arithmetic function template returns Tau(n), the number of divisors of n ($\scriptstyle \tau (n)\,$) of a nonzero integer, otherwise returns an error message.

Usage

{{tau|a nonzero integer}}

or

{{number of divisors|a nonzero integer}}

Valid input

A nonzero integer less than 1031 2 = 1062961 (validation is done by the {{mpf}} arithmetic function template).

Examples

Examples with valid input (check with https://oeis.org/A000005/b000005.txt Table of n, tau(n) for n = 1..100000)

Unfortunately, with the transclusion of {{number of divisors/doc}} via the {{documentation}} template the precious limited nesting levels of templates and/or parser functions were exhausted! :-( Check {{number of divisors/doc}} directly to see that all the tests are successful. Fortunately, by transcluding {{number of divisors/doc}} directly, borrowing the minimum code needed here from the {{documentation}} template, we manage to not exhaust the limit! :-)

Code	Result
{{number of divisors|210^2}}	81
{{tau|210^2}}	81
{{tau|-28}}	6
{{tau|-5}}	2
{{tau|1}}	1
{{tau|7}}	2
{{tau|15}}	4
{{tau|27}}	4
{{tau|30}}	8
{{tau|111}}	4
{{tau|5^3 * 11^2}}	12
{{tau|2^5 * 3^3 * 5}}	48
{{tau|2^9 * 3^3}}	40
{{tau|37^2 + 8 * 37^2}}	9
{{tau|2^9 * (26 + 1)}}	40
{{tau|89 * 113}}	4
{{tau|79 * 79}}	3
{{tau|210^2}}	81
{{tau|233^2}}	3
{{tau|10000}}	25
{{tau|65535}}	16
{{tau|65536}}	17
{{tau|65537}}	2
{{tau|65539}}	2
{{tau|65540}}	24
{{tau|65541}}	8
{{tau|65542}}	4
{{tau|65543}}	2
{{tau|65547}}	6
{{tau|65549}}	8
{{tau|65551}}	2
{{tau|65553}}	4
{{tau|65557}}	2
{{tau|65559}}	12
{{tau|65561}}	4
{{tau|65563}}	2
{{tau|65567}}	4
{{tau|65569}}	16
{{tau|65571}}	8
{{tau|65573}}	4
{{tau|65577}}	4
{{tau|65579}}	2
{{tau|265535}}	8
{{tau|265536}}	42
{{tau|265537}}	4
{{tau|265539}}	4
{{tau|265540}}	48
{{tau|265541}}	2
{{tau|265542}}	8
{{tau|265543}}	2
{{tau|265547}}	2
{{tau|265549}}	4
{{tau|265551}}	16
{{tau|265553}}	4
{{tau|265557}}	16
{{tau|265559}}	8
{{tau|265561}}	2
{{tau|265563}}	12
{{tau|265567}}	2
{{tau|265569}}	4
{{tau|265571}}	2
{{tau|265573}}	8
{{tau|265577}}	8
{{tau|265579}}	2
{{tau|257}}	2
{{tau|97 * 211}}	4
{{tau|216 * 211}}	32
{{tau|1024 * 45}}	66
{{tau|97 * 257}}	4
{{tau|3^6 * 5^2}}	21
{{tau|3 * 5^5}}	12
{{tau|17^2 * 191}}	6
{{tau|5 * 7 * 13 * 29}}	16
{{tau|509^2}}	3
{{tau|965535}}	16
{{tau|965536}}	48
{{tau|965537}}	4
{{tau|965539}}	4
{{tau|965540}}	24
{{tau|965541}}	4
{{tau|965542}}	8
{{tau|965543}}	4
{{tau|965547}}	16
{{tau|965549}}	12
{{tau|965551}}	2
{{tau|965553}}	4
{{tau|965557}}	4
{{tau|965559}}	8
{{tau|965561}}	8
{{tau|965563}}	4
{{tau|965567}}	2
{{tau|965569}}	8
{{tau|965571}}	8
{{tau|965573}}	8
{{tau|965577}}	8
{{tau|965579}}	4
{{tau|1015941}}	8
{{tau|997 * 1019}}	4
{{tau|1015943}}	4
{{tau|1015945}}	8
{{tau|1015947}}	12
{{tau|1015949}}	8
{{tau|1015950}}	48

Examples with invalid input (argument validation by {{number of divisors}} is omitted to spare some precious limited nesting levels of templates and/or parser functions).

Code	Result
{{tau|0}}
{{tau|1031^2}}

Code

<noinclude><!-- {{documentation}} --><!-- We can't use it here, we reached the nesting levels limit of templates and/or parser functions! 

So we just borrow the necessary code from it instead.

--><div style="text-align: center; font-size: smaller;">The following [[Help:Documenting templates|documentation]] is located at [[Template:{{PAGENAME}}/doc]].</div>{{Template:{{PAGENAME}}/doc}}<!--

--></noinclude><includeonly>{{#expr: (1 + 0{{mpf| {{{1|1}}} |sep = ) * (1 + |key/val_sep = ^0 * }} ) }}</includeonly>

See also

• {{if prime}} or {{is prime}}

• {{distinct prime factors up to sqrt(n)}} or {{dpf le sqrt(n)}}
• {{distinct nontrivial prime factors}} or {{dpf lt n}}
• {{distinct prime factors}} or {{dpf}}
• {{number of distinct prime factors}} or {{little omega}}
• {{sum of distinct prime factors}} or {{sodpf}}
• {{product of distinct prime factors}} or {{squarefree kernel}} or {{radical}} or {{rad}}

• {{multiplicity}}

• {{prime factors (with multiplicity) up to sqrt(n)}} or {{mpf le sqrt(n)}}
• {{nontrivial prime factors (with multiplicity)}} or {{mpf lt n}}
• {{prime factors (with multiplicity)}} or {{mpf}} or {{factorization}}
• {{number of prime factors (with multiplicity)}} or {{big Omega}}
• {{sum of prime factors (with multiplicity)}} or {{sopfr}} or {{integer log}}
• {{product of prime factors (with multiplicity)}} (must give back {{abs|n}}, the absolute value of

  n

  )

• {{quadratfrei}}
• {{Moebius mu}} or {{mu}}

• {{Euler phi}} or {{totient}}
• {{Dedekind psi}}

• {{number of divisors}} or {{sigma 0}} or {{tau}}
• {{sum of divisors}} or {{sigma 1}} or {{sigma}} (Cf. {{divisor function}} or {{sigma k}}, with

  k = 1

  (default value))
• {{divisor function}} or {{sigma k}} (for

  k   ≠   0

  )

External links

• Andrew Hodges, Java Applet for Factorization
• http://factordb.com/