Template:Moebius mu/doc

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[⧼Purge⧽ Template:Moebius mu/doc]

This template is under construction.            

Please do not use this unfinished and/or still unreliable template.            

Warning: All templates/parser functions nesting levels are used, so you can't nest it any further (e.g. you can't transclude the {{Moebius mu/doc}} subpage) without causing deeply nested templates to fail to render, giving wrong results. There is also the minor (though nagging) issue of minus sign before 0 when 0 is multiplied by a negative (-1)^(little omega).

The {{Moebius mu}} ({{mu}}) arithmetic function template returns the Moebius (or Mobius) function mu(n), $\scriptstyle \mu (n),\,$ of a nonzero integer, otherwise returns an error message.

Usage

{{Moebius mu|a nonzero integer}}

or

{{mu|a nonzero integer}}

Valid input

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

Examples

Examples with valid input (check with https://oeis.org/A008683/b008683.txt Table of n, mu(n) for n = 1..100000)

Unfortunately, with the transclusion of {{Moebius mu/doc}} via the {{documentation}} template the precious limited nesting levels of templates and/or parser functions were exhausted! :-( Check {{Moebius mu/doc}} directly to see that all the tests are successful. Unfortunately, by transcluding {{Moebius mu/doc}} directly, borrowing the minimum code needed here from the {{documentation}} template, we still exhaust the limit! :-(

Because the current code is inefficient (it calls the {{mpf}} template twice, and to avoid this we need to use a core function template to which we pass the result of mpf to some argument, problem is we have no spare precious limited templates/parser functions nesting levels!)

Code	Result
{{Moebius mu|210^2}}	0
{{mu|210^2}}	0
{{mu|-28}}	0
{{mu|-5}}	-1
{{mu|1}}	1
{{mu|7}}	-1
{{mu|15}}	1
{{mu|27}}	-0
{{mu|30}}	-1
{{mu|111}}	1
{{mu|5^3 * 11^2}}	0
{{mu|2^5 * 3^3 * 5}}	-0
{{mu|2^9 * 3^3}}	0
{{mu|37^2 + 8 * 37^2}}	0
{{mu|2^9 * (26 + 1)}}	0
{{mu|89 * 113}}	1
{{mu|79 * 79}}	-0
{{mu|210^2}}	0
{{mu|233^2}}	-0
{{mu|10000}}	0
{{mu|15535}}	-1
{{mu|15536}}	0
{{mu|15537}}	1
{{mu|15539}}	1
{{mu|15540}}	-0
{{mu|15541}}	-1
{{mu|15542}}	-1
{{mu|15543}}	-0
{{mu|15547}}	1
{{mu|15549}}	-1
{{mu|15551}}	-1
{{mu|15553}}	1
{{mu|15557}}	1
{{mu|15559}}	-1
{{mu|15561}}	0
{{mu|15563}}	1
{{mu|15567}}	1
{{mu|15569}}	-1
{{mu|15571}}	1
{{mu|15573}}	-1
{{mu|15577}}	1
{{mu|15579}}	0
{{mu|65535}}	1
{{mu|65536}}	-0
{{mu|65537}}	-1
{{mu|65539}}	-1
{{mu|65540}}	0
{{mu|65541}}	-1
{{mu|65542}}	1
{{mu|65543}}	-1
{{mu|65547}}	0
{{mu|65549}}	-1
{{mu|65551}}	-1
{{mu|65553}}	1
{{mu|65557}}	-1
{{mu|65559}}	-0
{{mu|65561}}	1
{{mu|65563}}	-1
{{mu|65567}}	1
{{mu|65569}}	1
{{mu|65571}}	-1
{{mu|65573}}	1
{{mu|65577}}	1
{{mu|65579}}	-1
{{mu|95561}}	-1
{{mu|95563}}	1
{{mu|95567}}	1
{{mu|95569}}	-1
{{mu|95571}}	0
{{mu|95573}}	1
{{mu|95577}}	1
{{mu|95579}}	1
{{mu|965561}}	-1
{{mu|965563}}	1
{{mu|965567}}	-1
{{mu|965569}}	-1
{{mu|965571}}	-1
{{mu|965573}}	-1
{{mu|965577}}	-1
{{mu|965579}}	1
{{mu|1015941}}	-1
{{mu|997 * 1019}}	1
{{mu|1015943}}	1
{{mu|1015945}}	-1
{{mu|1015947}}	-0
{{mu|1015949}}	-1
{{mu|1015950}}	-0

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

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

Code

<noinclude><!-- {{documentation}} --><!-- We can't use it here, the precious limited nesting levels of templates and/or parser functions get exhausted! 

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}}<!--

Inefficient: we call the mpf template twice.  To avoid this, we have to use a core function template where we pass the result of the mpf template call as some argument, problem is we don't have any spare of those precious limited templates/parser functions nesting levels! 

--></noinclude><includeonly>{{#expr: 
<!-- quadratfrei -->( not ( ( abs ({{{1|1}}}) ) - ( 0{{mpf| {{{1|1}}} |sep = * |key/val_sep = * 1^}} + {{#ifexpr: abs ({{{1|1}}}) = 1 | 1 | 0 }} ) ) ) 
* 
<!-- (-1)^(little omega) -->(-1)^( 0{{mpf| {{{1|1}}} |sep = + |key/val_sep = ^0 * 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/