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Template:Dedekind psi/doc
From OeisWiki
The {{Dedekind psi}} arithmetic function template returns the Dedekind psi function of () of a nonzero integer, otherwise returns an error message.
Usage
- {{Dedekind psi|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/A001615/b001615.txt Table of n, psi(n) for n = 1..10000)
Unfortunately, with the transclusion of {{Dedekind psi/doc}} via the {{documentation}} template the precious limited nesting levels of templates and/or parser functions were exhausted! :-( Check {{Dedekind psi/doc}} directly to see that all the tests are successful. Fortunately, by transcluding {{Dedekind psi/doc}} directly, borrowing the minimum code needed here from the {{documentation}} template, we manage to not exhaust the limit! :-)
Code Result {{Dedekind psi|210^2}} 120960 {{Dedekind psi|210^2}} 120960 {{Dedekind psi|-28}} 48 {{Dedekind psi|-5}} 6 {{Dedekind psi|1}} 1 {{Dedekind psi|7}} 8 {{Dedekind psi|15}} 24 {{Dedekind psi|27}} 36 {{Dedekind psi|30}} 72 {{Dedekind psi|111}} 152 {{Dedekind psi|5^3 * 11^2}} 19800 {{Dedekind psi|2^5 * 3^3 * 5}} 10368 {{Dedekind psi|2^9 * 3^3}} 27648 {{Dedekind psi|37^2 + 8 * 37^2}} 16872 {{Dedekind psi|2^9 * (26 + 1)}} 27648 {{Dedekind psi|89 * 113}} 10260 {{Dedekind psi|79 * 79}} 6320 {{Dedekind psi|210^2}} 120960 {{Dedekind psi|233^2}} 54522 {{Dedekind psi|10000}} 18000 {{Dedekind psi|65535}} 111456 {{Dedekind psi|65536}} 98304 {{Dedekind psi|65537}} 65538 {{Dedekind psi|65539}} 65540 {{Dedekind psi|65540}} 123120 {{Dedekind psi|65541}} 99904 {{Dedekind psi|65542}} 98316 {{Dedekind psi|65543}} 65544 {{Dedekind psi|65547}} 87408 {{Dedekind psi|65549}} 73440 {{Dedekind psi|65551}} 65552 {{Dedekind psi|65553}} 87408 {{Dedekind psi|65557}} 65558 {{Dedekind psi|65559}} 96432 {{Dedekind psi|65561}} 66852 {{Dedekind psi|65563}} 65564 {{Dedekind psi|65567}} 66120 {{Dedekind psi|65569}} 86400 {{Dedekind psi|65571}} 95424 {{Dedekind psi|65573}} 68448 {{Dedekind psi|65577}} 87440 {{Dedekind psi|65579}} 65580 {{Dedekind psi|265535}} 332640 {{Dedekind psi|265536}} 532224 {{Dedekind psi|265537}} 267696 {{Dedekind psi|265539}} 354056 {{Dedekind psi|265540}} 559872 {{Dedekind psi|265541}} 265542 {{Dedekind psi|265542}} 531096 {{Dedekind psi|265543}} 265544 {{Dedekind psi|265547}} 265548 {{Dedekind psi|265549}} 272764 {{Dedekind psi|265551}} 416640 {{Dedekind psi|265553}} 274740 {{Dedekind psi|265557}} 387072 {{Dedekind psi|265559}} 309120 {{Dedekind psi|265561}} 265562 {{Dedekind psi|265563}} 372960 {{Dedekind psi|265567}} 265568 {{Dedekind psi|265569}} 354096 {{Dedekind psi|265571}} 265572 {{Dedekind psi|265573}} 331200 {{Dedekind psi|265577}} 295680 {{Dedekind psi|265579}} 265580 {{Dedekind psi|257}} 258 {{Dedekind psi|97 * 211}} 20776 {{Dedekind psi|216 * 211}} 91584 {{Dedekind psi|1024 * 45}} 110592 {{Dedekind psi|97 * 257}} 25284 {{Dedekind psi|3^6 * 5^2}} 29160 {{Dedekind psi|3 * 5^5}} 15000 {{Dedekind psi|17^2 * 191}} 58752 {{Dedekind psi|5 * 7 * 13 * 29}} 20160 {{Dedekind psi|509^2}} 259590 {{Dedekind psi|965535}} 1572480 {{Dedekind psi|965536}} 1709568 {{Dedekind psi|965537}} 980016 {{Dedekind psi|965539}} 977256 {{Dedekind psi|965540}} 1814400 {{Dedekind psi|965541}} 1287392 {{Dedekind psi|965542}} 1524600 {{Dedekind psi|965543}} 968448 {{Dedekind psi|965547}} 1404864 {{Dedekind psi|965549}} 1105272 {{Dedekind psi|965551}} 965552 {{Dedekind psi|965553}} 1287408 {{Dedekind psi|965557}} 996736 {{Dedekind psi|965559}} 1471360 {{Dedekind psi|965561}} 1029600 {{Dedekind psi|965563}} 1007568 {{Dedekind psi|965567}} 965568 {{Dedekind psi|965569}} 1071360 {{Dedekind psi|965571}} 1305360 {{Dedekind psi|965573}} 1109760 {{Dedekind psi|965577}} 1292416 {{Dedekind psi|965579}} 970776 {{Dedekind psi|1015941}} 1373440 {{Dedekind psi|997 * 1019}} 1017960 {{Dedekind psi|1015943}} 1017960 {{Dedekind psi|1015945}} 1393344 {{Dedekind psi|1015947}} 1365120 {{Dedekind psi|1015949}} 1166880 {{Dedekind psi|1015950}} 2630880
Examples with invalid input (argument validation by {{Dedekind psi}} is omitted to spare some precious limited nesting levels of templates and/or parser functions).
Code Result {{Dedekind psi|0}} Expression error: Unrecognized word "strong". {{Dedekind psi|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/
