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Template:Product of prime factors (with multiplicity)
The {{product of prime factors (with multiplicity)}} (which must give the same result as {{abs|n}}) arithmetic function template returns the product of prime factors of n (with multiplicity) (absolute value) of a nonzero integer, otherwise returns an error message.
Usage
- {{product of prime factors (with multiplicity)|a nonzero integer}}
must give the same result as
- {{abs|a nonzero integer}}
which simply returns the absolute value, if the {{prime factors (with multiplicity)}} ({{mpf}} or {{factorization}}) arithmetic function template worked properly (which means that not only it is not buggy, but also the precious limited nesting levels of templates and/or parser functions were NOT exhausted!).
Valid input
A nonzero integer less than 1031 2 = 1062961 (validation is done by the {{mpf}} arithmetic function template).
Examples
Examples with valid input
Unfortunately, with the transclusion of {{product of prime factors (with multiplicity)/doc}} via the {{documentation}} template the precious limited nesting levels of templates and/or parser functions were exhausted! :-( Check {{product of prime factors (with multiplicity)/doc}} directly to see that all the tests are successful. Fortunately, by transcluding {{product of prime factors (with multiplicity)/doc}} directly, borrowing the minimum code needed here from the {{documentation}} template, we manage to not exhaust the limit! :-)
Code Result Comment {{product of prime factors (with multiplicity)|210^2}} 44100 {{product of prime factors (with multiplicity)|-28}} 28 {{product of prime factors (with multiplicity)|-5}} 5 {{product of prime factors (with multiplicity)|-1}} 1 (empty product, i.e 1) {{product of prime factors (with multiplicity)|1}} 1 (empty product, i.e 1) {{product of prime factors (with multiplicity)|7}} 7 {{product of prime factors (with multiplicity)|15}} 15 {{product of prime factors (with multiplicity)|27}} 27 {{product of prime factors (with multiplicity)|30}} 30 {{product of prime factors (with multiplicity)|111}} 111 {{product of prime factors (with multiplicity)|5^3 * 11^2}} 15125 {{product of prime factors (with multiplicity)|2^5 * 3^3 * 5}} 4320 {{product of prime factors (with multiplicity)|2^9 * 3^3}} 13824 {{product of prime factors (with multiplicity)|37^2 + 8 * 37^2}} 12321 {{product of prime factors (with multiplicity)|2^9 * (26 + 1)}} 13824 {{product of prime factors (with multiplicity)|89 * 113}} 10057 {{product of prime factors (with multiplicity)|79 * 79}} 6241 {{product of prime factors (with multiplicity)|210^2}} 44100 {{product of prime factors (with multiplicity)|233^2}} 54289 {{product of prime factors (with multiplicity)|10000}} 10000 {{product of prime factors (with multiplicity)|65535}} 65535 {{product of prime factors (with multiplicity)|65536}} 65536 {{product of prime factors (with multiplicity)|65537}} 65537 {{product of prime factors (with multiplicity)|65539}} 65539 {{product of prime factors (with multiplicity)|65540}} 65540 {{product of prime factors (with multiplicity)|65541}} 65541 {{product of prime factors (with multiplicity)|65542}} 65542 {{product of prime factors (with multiplicity)|65543}} 65543 {{product of prime factors (with multiplicity)|65547}} 65547 {{product of prime factors (with multiplicity)|65549}} 65549 {{product of prime factors (with multiplicity)|65551}} 65551 {{product of prime factors (with multiplicity)|65553}} 65553 {{product of prime factors (with multiplicity)|65557}} 65557 {{product of prime factors (with multiplicity)|65559}} 65559 {{product of prime factors (with multiplicity)|65561}} 65561 {{product of prime factors (with multiplicity)|65563}} 65563 {{product of prime factors (with multiplicity)|65567}} 65567 {{product of prime factors (with multiplicity)|65569}} 65569 {{product of prime factors (with multiplicity)|65571}} 65571 {{product of prime factors (with multiplicity)|65573}} 65573 {{product of prime factors (with multiplicity)|65577}} 65577 {{product of prime factors (with multiplicity)|65579}} 65579 {{product of prime factors (with multiplicity)|265535}} 265535 {{product of prime factors (with multiplicity)|265536}} 265536 {{product of prime factors (with multiplicity)|265537}} 265537 {{product of prime factors (with multiplicity)|265539}} 265539 {{product of prime factors (with multiplicity)|265540}} 265540 {{product of prime factors (with multiplicity)|265541}} 265541 {{product of prime factors (with multiplicity)|265542}} 265542 {{product of prime factors (with multiplicity)|265543}} 265543 {{product of prime factors (with multiplicity)|265547}} 265547 {{product of prime factors (with multiplicity)|265549}} 265549 {{product of prime factors (with multiplicity)|265551}} 265551 {{product of prime factors (with multiplicity)|265553}} 265553 {{product of prime factors (with multiplicity)|265557}} 265557 {{product of prime factors (with multiplicity)|265559}} 265559 {{product of prime factors (with multiplicity)|265561}} 265561 {{product of prime factors (with multiplicity)|265563}} 265563 {{product of prime factors (with multiplicity)|265567}} 265567 {{product of prime factors (with multiplicity)|265569}} 265569 {{product of prime factors (with multiplicity)|265571}} 265571 {{product of prime factors (with multiplicity)|265573}} 265573 {{product of prime factors (with multiplicity)|265577}} 265577 {{product of prime factors (with multiplicity)|265579}} 265579 {{product of prime factors (with multiplicity)|257}} 257 {{product of prime factors (with multiplicity)|97 * 211}} 20467 {{product of prime factors (with multiplicity)|216 * 211}} 45576 {{product of prime factors (with multiplicity)|1024 * 45}} 46080 {{product of prime factors (with multiplicity)|97 * 257}} 24929 {{product of prime factors (with multiplicity)|3^6 * 5^2}} 18225 {{product of prime factors (with multiplicity)|3 * 5^5}} 9375 {{product of prime factors (with multiplicity)|17^2 * 191}} 55199 {{product of prime factors (with multiplicity)|5 * 7 * 13 * 29}} 13195 {{product of prime factors (with multiplicity)|509^2}} 259081 {{product of prime factors (with multiplicity)|965535}} 965535 {{product of prime factors (with multiplicity)|965536}} 965536 {{product of prime factors (with multiplicity)|965537}} 965537 {{product of prime factors (with multiplicity)|965539}} 965539 {{product of prime factors (with multiplicity)|965540}} 965540 {{product of prime factors (with multiplicity)|965541}} 965541 {{product of prime factors (with multiplicity)|965542}} 965542 {{product of prime factors (with multiplicity)|965543}} 965543 {{product of prime factors (with multiplicity)|965547}} 965547 {{product of prime factors (with multiplicity)|965549}} 965549 {{product of prime factors (with multiplicity)|965551}} 965551 {{product of prime factors (with multiplicity)|965553}} 965553 {{product of prime factors (with multiplicity)|965557}} 965557 {{product of prime factors (with multiplicity)|965559}} 965559 {{product of prime factors (with multiplicity)|965561}} 965561 {{product of prime factors (with multiplicity)|965563}} 965563 {{product of prime factors (with multiplicity)|965567}} 965567 {{product of prime factors (with multiplicity)|965569}} 965569 {{product of prime factors (with multiplicity)|965571}} 965571 {{product of prime factors (with multiplicity)|965573}} 965573 {{product of prime factors (with multiplicity)|965577}} 965577 {{product of prime factors (with multiplicity)|965579}} 965579 {{product of prime factors (with multiplicity)|1015941}} 1015941 {{product of prime factors (with multiplicity)|997 * 1019}} 1015943 {{product of prime factors (with multiplicity)|1015943}} 1015943 {{product of prime factors (with multiplicity)|1015945}} 1015945 {{product of prime factors (with multiplicity)|1015947}} 1015947 {{product of prime factors (with multiplicity)|1015949}} 1015949 {{product of prime factors (with multiplicity)|1015950}} 1015950
Examples with invalid input (argument validation by {{product of prime factors (with multiplicity)}} is omitted to spare some precious limited nesting levels of templates and/or parser functions).
Code Result {{product of prime factors (with multiplicity)|0}} Expression error: Unrecognized word "strong". {{product of prime factors (with multiplicity)|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 = ^ }} + {{#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/