Template:Divisor function/doc

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The {{divisor function}} arithmetic function template returns σk (n) for

k   ≠   0

of a nonzero integer, otherwise returns an error message. For

k = 0

, see {{number of divisors}} or {{sigma 0}} or {{tau}} arithmetic function template.

Usage

{{sigma k|a nonzero integer|k = a nonzero integer (default 1)}}

or

{{divisor function|a nonzero integer|k = a nonzero integer (default 1)}}

or

{{sigma k|a nonzero integer}}

or

{{divisor function|a nonzero integer}}

and for

k = 0

use

{{number of divisors|a nonzero integer}} or {{sigma 0|a nonzero integer}} or {{tau|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 for

k   ≥   1

with Divisor function#Table of sequences)

Divisor function sequences

k	A-number
0	A000005  (n)
1	A000203  (n)
2	A001157  (n)
3	A001158  (n)
4	A001159  (n)
5	A001160  (n)
6	A013954  (n)
7	A013955  (n)
8	A013956  (n)

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
{{divisor function|24|k = 3}}	16380
{{divisor function|24|k = 2}}	850
{{divisor function|24|k = 1}}	60
{{divisor function|24}}	60
{{divisor function|210^2}}	160797
{{divisor function|210^2}}	160797
{{divisor function|-28}}	56
{{divisor function|-5}}	6
{{divisor function|1}}	1
{{divisor function|7}}	8
{{divisor function|15}}	24
{{divisor function|27}}	40
{{divisor function|30}}	72
{{divisor function|111}}	152
{{sigma k|5^3 * 11^2}}	20748
{{sigma k|2^5 * 3^3 * 5}}	15120
{{sigma k|2^9 * 3^3}}	40920
{{sigma k|37^2 + 8 * 37^2}}	18291
{{sigma k|2^9 * (26 + 1)}}	40920
{{sigma k|89 * 113}}	10260
{{sigma k|79 * 79}}	6321
{{sigma k|210^2}}	160797
{{sigma k|233^2}}	54523
{{sigma k|10000}}	24211
{{sigma k|65535}}	111456
{{sigma k|65536}}	131071
{{sigma k|65537}}	65538
{{sigma k|65539}}	65540
{{sigma k|65540}}	143640
{{sigma k|65541}}	99904
{{sigma k|65542}}	98316
{{sigma k|65543}}	65544
{{sigma k|65547}}	94692
{{sigma k|65549}}	73440
{{sigma k|65551}}	65552
{{sigma k|65553}}	87408
{{sigma k|65557}}	65558
{{sigma k|65559}}	96488
{{sigma k|65561}}	66852
{{sigma k|65563}}	65564
{{sigma k|65567}}	66120
{{sigma k|65569}}	86400
{{sigma k|65571}}	95424
{{sigma k|65573}}	68448
{{sigma k|65577}}	87440
{{sigma k|65579}}	65580

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

Code	Result
{{sigma k|0}}
{{sigma k|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}}<!--

:<math> \sigma_k(n) = \begin{cases}
\prod_{i=1}^{\omega(n)} (1 + \alpha_i) & \text{if } k = 0, \\
\prod_{i=1}^{\omega(n)} \frac{p_{i}^{k (1+ \alpha_i)} - 1}{p_{i}^{k} - 1} & \text{if } k > 0. \end{cases}</math>

For k = 0, we have a separate arithmetic function template: [[Template:Number of divisors]]

The following code is very inefficient, we call [[Template:mpf]] twice! To avoid this we would need a core function template to which we pass the result of [[Template:mpf]] to one of its arguments, but this requires one more templates and/or parser functions nesting level, unfortunately we've reached the limit! :-(

For -1 or +1, [[Template:mpf]] returns the empty string, so we use an ugly kludge (without adding a nesting level around [[Template:mpf]]) to get 1 as result, and to balance parenthesis. I admit that templates are no scripting language, and using templates to implement arithmetic functions is like using a screwdriver to hammer a nail...  

--></noinclude><includeonly>{{#expr: 
( 
  ( {{#ifexpr: abs ({{{1|1}}}) = 1 | ( ( 2 }}{{mpf| {{{1|1}}} |sep = ) ) - 1) * ( |key/val_sep = ^( ({{{k|{{{2|1}}}}}}) * (1 + }} ) ) - 1) 
) 
/ 
( 
  ( {{#ifexpr: abs ({{{1|1}}}) = 1 | ( 2 }}{{mpf| {{{1|1}}} |sep = ) - 1) * ( |key/val_sep = ^( ({{{k|{{{2|1}}}}}}) * 1^ }} ) - 1) 
) 
}}</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/