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Template:Number of divisors/doc
The {{number of divisors}} arithmetic function template returns Tau(n), the number of divisors of 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}} Expression error: Unrecognized word "strong". {{tau|1031^2}} Expression error: Unrecognized word "strong".
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
- {{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
(default value))k = 1 - {{divisor function}} or {{sigma k}} (for
)k ≠ 0
External links
- Andrew Hodges, Java Applet for Factorization
- http://factordb.com/