Template:Dedekind psi

The following documentation is located at Template:Dedekind psi/doc.

The {{Dedekind psi}} arithmetic function template returns the Dedekind psi function of $\scriptstyle n\,$ ($\scriptstyle \psi (n)\,$) 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}}
{{Dedekind psi|1031^2}}

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

• {{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/