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Template:Totient
From OeisWiki
The following documentation is located at Template:Totient/doc.
The {{totient}} arithmetic function template returns Euler's totient function of (Euler's phi function of , ) of a nonzero integer, otherwise returns an error message.
Usage
- {{totient|a nonzero integer}}
or
- {{Euler phi|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/A000010/b000010.txt Table of n, phi(n) for n = 1..100000)
Unfortunately, with the transclusion of {{totient/doc}} via the {{documentation}} template the precious limited nesting levels of templates and/or parser functions were exhausted! :-( Check {{totient/doc}} directly to see that all the tests are successful. Fortunately, by transcluding {{totient/doc}} directly, borrowing the minimum code needed here from the {{documentation}} template, we manage to not exhaust the limit! :-)
Code Result {{Euler phi|210^2}} 10080 {{totient|210^2}} 10080 {{totient|-28}} 12 {{totient|-5}} 4 {{totient|1}} 1 {{totient|7}} 6 {{totient|15}} 8 {{totient|27}} 18 {{totient|30}} 8 {{totient|111}} 72 {{totient|5^3 * 11^2}} 11000 {{totient|2^5 * 3^3 * 5}} 1152 {{totient|2^9 * 3^3}} 4608 {{totient|37^2 + 8 * 37^2}} 7992 {{totient|2^9 * (26 + 1)}} 4608 {{totient|89 * 113}} 9856 {{totient|79 * 79}} 6162 {{totient|210^2}} 10080 {{totient|233^2}} 54056 {{totient|10000}} 4000 {{totient|65535}} 32768 {{totient|65536}} 32768 {{totient|65537}} 65536 {{totient|65539}} 65538 {{totient|65540}} 25088 {{totient|65541}} 37440 {{totient|65542}} 32770 {{totient|65543}} 65542 {{totient|65547}} 43692 {{totient|65549}} 58000 {{totient|65551}} 65550 {{totient|65553}} 43700 {{totient|65557}} 65556 {{totient|65559}} 39360 {{totient|65561}} 64272 {{totient|65563}} 65562 {{totient|65567}} 65016 {{totient|65569}} 48384 {{totient|65571}} 39720 {{totient|65573}} 62700 {{totient|65577}} 43716 {{totient|65579}} 65578 {{totient|265535}} 203104 {{totient|265536}} 88320 {{totient|265537}} 263380 {{totient|265539}} 177024 {{totient|265540}} 89600 {{totient|265541}} 265540 {{totient|265542}} 88512 {{totient|265543}} 265542 {{totient|265547}} 265546 {{totient|265549}} 258336 {{totient|265551}} 148320 {{totient|265553}} 256368 {{totient|265557}} 161280 {{totient|265559}} 223416 {{totient|265561}} 265560 {{totient|265563}} 167616 {{totient|265567}} 265566 {{totient|265569}} 177044 {{totient|265571}} 265570 {{totient|265573}} 206880 {{totient|265577}} 236880 {{totient|265579}} 265578 {{totient|257}} 256 {{totient|97 * 211}} 20160 {{totient|216 * 211}} 15120 {{totient|1024 * 45}} 12288 {{totient|97 * 257}} 24576 {{totient|3^6 * 5^2}} 9720 {{totient|3 * 5^5}} 5000 {{totient|17^2 * 191}} 51680 {{totient|5 * 7 * 13 * 29}} 8064 {{totient|509^2}} 258572 {{totient|965535}} 505760 {{totient|965536}} 403200 {{totient|965537}} 951060 {{totient|965539}} 953824 {{totient|965540}} 369248 {{totient|965541}} 643692 {{totient|965542}} 457344 {{totient|965543}} 962640 {{totient|965547}} 585000 {{totient|965549}} 835584 {{totient|965551}} 965550 {{totient|965553}} 643700 {{totient|965557}} 934380 {{totient|965559}} 551736 {{totient|965561}} 902880 {{totient|965563}} 923560 {{totient|965567}} 965566 {{totient|965569}} 862800 {{totient|965571}} 634752 {{totient|965573}} 822960 {{totient|965577}} 641232 {{totient|965579}} 960384 {{totient|1015941}} 667872 {{totient|997 * 1019}} 1013928 {{totient|1015943}} 1013928 {{totient|1015945}} 696624 {{totient|1015947}} 672048 {{totient|1015949}} 874800 {{totient|1015950}} 249600
Examples with invalid input (argument validation by {{totient}} is omitted to spare some precious limited nesting levels of templates and/or parser functions).
Code Result {{totient|0}} Expression error: Unrecognized word "strong". {{totient|1031^2}} Expression error: Unrecognized word "strong".
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}}<!--
--></noinclude><includeonly>{{#expr: ( abs ({{{1|1}}}) ) * (1 - 0{{mpf| {{{1|1}}} |sep = ) * (1 - |key/val_sep = ^(-1) * 1^}} ) }}</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/
