Template:Quadratfrei

The following documentation is located at Template:Quadratfrei/doc.

The {{quadratfrei}} arithmetic function template returns the characteristic function of squarefree numbers of $\scriptstyle n\,$ (quadratfrei function of $\scriptstyle n\,$, $\scriptstyle q(n)\,$) of a nonzero integer, otherwise returns an error message.

Usage

{{quadratfrei|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/A008966/b008966.txt Table of n, rad(n) for n = 1..100000)

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

Code	Result
{{quadratfrei|210^2}}	0
{{quadratfrei|-28}}	0
{{quadratfrei|-5}}	1
{{quadratfrei|1}}	1
{{quadratfrei|7}}	1
{{quadratfrei|15}}	1
{{quadratfrei|27}}	0
{{quadratfrei|30}}	1
{{quadratfrei|111}}	1
{{quadratfrei|5^3 * 11^2}}	0
{{quadratfrei|2^5 * 3^3 * 5}}	0
{{quadratfrei|2^9 * 3^3}}	0
{{quadratfrei|37^2 + 8 * 37^2}}	0
{{quadratfrei|2^9 * (26 + 1)}}	0
{{quadratfrei|89 * 113}}	1
{{quadratfrei|79 * 79}}	0
{{quadratfrei|210^2}}	0
{{quadratfrei|233^2}}	0
{{quadratfrei|10000}}	0
{{quadratfrei|65535}}	1
{{quadratfrei|65536}}	0
{{quadratfrei|65537}}	1
{{quadratfrei|65539}}	1
{{quadratfrei|65540}}	0
{{quadratfrei|65541}}	1
{{quadratfrei|65542}}	1
{{quadratfrei|65543}}	1
{{quadratfrei|65547}}	0
{{quadratfrei|65549}}	1
{{quadratfrei|65551}}	1
{{quadratfrei|65553}}	1
{{quadratfrei|65557}}	1
{{quadratfrei|65559}}	0
{{quadratfrei|65561}}	1
{{quadratfrei|65563}}	1
{{quadratfrei|65567}}	1
{{quadratfrei|65569}}	1
{{quadratfrei|65571}}	1
{{quadratfrei|65573}}	1
{{quadratfrei|65577}}	1
{{quadratfrei|65579}}	1
{{quadratfrei|265535}}	1
{{quadratfrei|265536}}	0
{{quadratfrei|265537}}	1
{{quadratfrei|265539}}	1
{{quadratfrei|265540}}	0
{{quadratfrei|265541}}	1
{{quadratfrei|265542}}	1
{{quadratfrei|265543}}	1
{{quadratfrei|265547}}	1
{{quadratfrei|265549}}	1
{{quadratfrei|265551}}	1
{{quadratfrei|265553}}	1
{{quadratfrei|265557}}	1
{{quadratfrei|265559}}	1
{{quadratfrei|265561}}	1
{{quadratfrei|265563}}	0
{{quadratfrei|265567}}	1
{{quadratfrei|265569}}	1
{{quadratfrei|265571}}	1
{{quadratfrei|265573}}	1
{{quadratfrei|265577}}	1
{{quadratfrei|265579}}	1
{{quadratfrei|257}}	1
{{quadratfrei|97 * 211}}	1
{{quadratfrei|216 * 211}}	0
{{quadratfrei|1024 * 45}}	0
{{quadratfrei|97 * 257}}	1
{{quadratfrei|3^6 * 5^2}}	0
{{quadratfrei|3 * 5^5}}	0
{{quadratfrei|17^2 * 191}}	0
{{quadratfrei|5 * 7 * 13 * 29}}	1
{{quadratfrei|509^2}}	0
{{quadratfrei|965535}}	1
{{quadratfrei|965536}}	0
{{quadratfrei|965537}}	1
{{quadratfrei|965539}}	1
{{quadratfrei|965540}}	0
{{quadratfrei|965541}}	1
{{quadratfrei|965542}}	1
{{quadratfrei|965543}}	1
{{quadratfrei|965547}}	0
{{quadratfrei|965549}}	0
{{quadratfrei|965551}}	1
{{quadratfrei|965553}}	1
{{quadratfrei|965557}}	1
{{quadratfrei|965559}}	1
{{quadratfrei|965561}}	1
{{quadratfrei|965563}}	1
{{quadratfrei|965567}}	1
{{quadratfrei|965569}}	1
{{quadratfrei|965571}}	1
{{quadratfrei|965573}}	1
{{quadratfrei|965577}}	1
{{quadratfrei|965579}}	1
{{quadratfrei|1015941}}	1
{{quadratfrei|997 * 1019}}	1
{{quadratfrei|1015943}}	1
{{quadratfrei|1015945}}	1
{{quadratfrei|1015947}}	0
{{quadratfrei|1015949}}	1
{{quadratfrei|1015950}}	0

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

Code	Result
{{quadratfrei|0}}
{{quadratfrei|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}}<!--

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