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Template:Product of distinct prime factors/doc
The {{product of distinct prime factors}} arithmetic function template returns the product of distinct prime factors of n (squarefree kernel of n) (radical of n, rad(n)) of a nonzero integer, otherwise returns an error message.
Usage
- {{product of distinct prime factors|a nonzero integer}}
or
- {{squarefree kernel|a nonzero integer}}
or
- {{radical|a nonzero integer}}
or
- {{rad|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/A007947/b007947.txt Table of n, rad(n) for n = 1..100000)
Unfortunately, with the transclusion of {{product of distinct prime factors/doc}} via the {{documentation}} template the precious limited nesting levels of templates and/or parser functions were exhausted! :-( Check {{product of distinct prime factors/doc}} directly to see that all the tests are successful. Fortunately, by transcluding {{product of distinct prime factors/doc}} directly, borrowing the minimum code needed here from the {{documentation}} template, we manage to not exhaust the limit! :-)
Code Result Comment {{product of distinct prime factors|210^2}} 210 {{squarefree kernel|210^2}} 210 {{radical|210^2}} 210 {{rad|210^2}} 210 {{rad|-28}} 14 {{rad|-5}} 5 {{rad|-1}} 1 (empty product, i.e 1) {{rad|1}} 1 (empty product, i.e 1) {{rad|7}} 7 {{rad|15}} 15 {{rad|27}} 3 {{rad|30}} 30 {{rad|111}} 111 {{rad|5^3 * 11^2}} 55 {{rad|2^5 * 3^3 * 5}} 30 {{rad|2^9 * 3^3}} 6 {{rad|37^2 + 8 * 37^2}} 111 {{rad|2^9 * (26 + 1)}} 6 {{rad|89 * 113}} 10057 {{rad|79 * 79}} 79 {{rad|210^2}} 210 {{rad|233^2}} 233 {{rad|10000}} 10 {{rad|65535}} 65535 {{rad|65536}} 2 {{rad|65537}} 65537 {{rad|65539}} 65539 {{rad|65540}} 32770 {{rad|65541}} 65541 {{rad|65542}} 65542 {{rad|65543}} 65543 {{rad|65547}} 21849 {{rad|65549}} 65549 {{rad|65551}} 65551 {{rad|65553}} 65553 {{rad|65557}} 65557 {{rad|65559}} 1599 {{rad|65561}} 65561 {{rad|65563}} 65563 {{rad|65567}} 65567 {{rad|65569}} 65569 {{rad|65571}} 65571 {{rad|65573}} 65573 {{rad|65577}} 65577 {{rad|65579}} 65579 {{rad|265535}} 265535 {{rad|265536}} 2766 {{rad|265537}} 265537 {{rad|265539}} 265539 {{rad|265540}} 132770 {{rad|265541}} 265541 {{rad|265542}} 265542 {{rad|265543}} 265543 {{rad|265547}} 265547 {{rad|265549}} 265549 {{rad|265551}} 265551 {{rad|265553}} 265553 {{rad|265557}} 265557 {{rad|265559}} 265559 {{rad|265561}} 265561 {{rad|265563}} 88521 {{rad|265567}} 265567 {{rad|265569}} 265569 {{rad|265571}} 265571 {{rad|265573}} 265573 {{rad|265577}} 265577 {{rad|265579}} 265579 {{rad|257}} 257 {{rad|97 * 211}} 20467 {{rad|216 * 211}} 1266 {{rad|1024 * 45}} 30 {{rad|97 * 257}} 24929 {{rad|3^6 * 5^2}} 15 {{rad|3 * 5^5}} 15 {{rad|17^2 * 191}} 3247 {{rad|5 * 7 * 13 * 29}} 13195 {{rad|509^2}} 509 {{rad|965535}} 965535 {{rad|965536}} 60346 {{rad|965537}} 965537 {{rad|965539}} 965539 {{rad|965540}} 482770 {{rad|965541}} 965541 {{rad|965542}} 965542 {{rad|965543}} 965543 {{rad|965547}} 107283 {{rad|965549}} 56797 {{rad|965551}} 965551 {{rad|965553}} 965553 {{rad|965557}} 965557 {{rad|965559}} 965559 {{rad|965561}} 965561 {{rad|965563}} 965563 {{rad|965567}} 965567 {{rad|965569}} 965569 {{rad|965571}} 965571 {{rad|965573}} 965573 {{rad|965577}} 965577 {{rad|965579}} 965579 {{rad|1015941}} 1015941 {{rad|997 * 1019}} 1015943 {{rad|1015943}} 1015943 {{rad|1015945}} 1015945 {{rad|1015947}} 338649 {{rad|1015949}} 1015949 {{rad|1015950}} 203190
Examples with invalid input (argument validation by {{rad}} is omitted to spare some precious limited nesting levels of templates and/or parser functions).
Code Result {{rad|0}} Expression error: Unrecognized word "strong". {{rad|1031^2}} Expression error: Unrecognized word "strong".
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: 0{{mpf| {{{1|1}}} |sep = * |key/val_sep = * 1^}} + {{#ifexpr: abs ({{{1|1}}}) = 1 | 1 | 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/
