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| In [[number theory]], a '''perfect totient number''' is an [[integer]] that is equal to the sum of its iterated [[totient]]s. That is, we apply the [[totient function]] to a number ''n'', apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals ''n'', then ''n'' is a perfect totient number. Or to put it algebraically, if
| | Hello and welcome. My name is Numbers Wunder. North Dakota is our beginning location. To play baseball is the hobby he will never quit performing. Supervising is my profession.<br><br>My web-site - [http://spermdonorinfo.co.uk/groups/eliminate-a-yeast-infection-with-one-of-these-tips/ home std test] |
| :<math>n = \sum_{i = 1}^{c + 1} \varphi^i(n),</math>
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| where
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| :<math>\varphi^i(n)=\left\{\begin{matrix}\varphi(n)&\mbox{ if } i=1\\ \varphi(\varphi^{i-1}(n))&\mbox{ otherwise}\end{matrix}\right.</math>
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| is the iterated totient function and ''c'' is the integer such that
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| :<math>\displaystyle\varphi^c(n)=2,</math>
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| then ''n'' is a perfect totient number.
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| The first few perfect totient numbers are
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| :[[3 (number)|3]], [[9 (number)|9]], [[15 (number)|15]], [[27 (number)|27]], [[39 (number)|39]], [[81 (number)|81]], [[111 (number)|111]], [[183 (number)|183]], [[243 (number)|243]], [[255 (number)|255]], [[327 (number)|327]], [[363 (number)|363]], 471, [[729 (number)|729]], 2187, 2199, 3063, 4359, 4375, ... {{OEIS|id=A082897}}.
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| For example, start with 327. Then φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327.
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| == Multiples and powers of three ==
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| It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that
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| :<math>\displaystyle\varphi(3^k) = \varphi(2\times 3^k) = 2\times 3^{k-1}.</math> | |
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| Venkataraman (1975) found another family of perfect totient numbers: if ''p'' = 4×3<sup>k</sup>+1 is prime, then 3''p'' is a perfect totient number. The values of ''k'' leading to perfect totient numbers in this way are
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| :0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... {{OEIS|id=A005537}}.
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| More generally if ''p'' is a [[prime number]] greater than 3, and 3''p'' is a perfect totient number, then ''p'' ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all ''p'' of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9''p'' is a perfect totient number then ''p'' is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3<sup>k</sup>''p'' where ''p'' is prime and ''k'' > 3.
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| ==References==
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| *{{cite journal
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| | author = Pérez-Cacho Villaverde, Laureano
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| | title = Sobre la suma de indicadores de ordenes sucesivos
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| | journal = Revista Matematica Hispano-Americana
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| | volume = 5
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| | issue = 3
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| | year = 1939
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| | pages = 45–50}}
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| *{{cite book
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| | author = Guy, Richard K.
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| | authorlink = Richard K. Guy
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| | title = Unsolved Problems in Number Theory
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| | location = New York
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| | publisher = Springer-Verlag
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| | year = 2004
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| | isbn=0-387-20860-7
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| | page = §B41}}
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| *{{cite journal
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| | author = Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L.
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| | title = On perfect totient numbers
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| | journal = [[Journal of Integer Sequences]]
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| | volume = 6
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| | year = 2003
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| | issue = 4
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| | pages = 03.4.5
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| | id = {{MathSciNet | id = 2051959}}
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| | url = http://www.emis.de/journals/JIS/VOL6/Cohen2/cohen50.pdf}}
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| *{{cite journal
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| | author = Luca, Florian
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| | title = On the distribution of perfect totients
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| | journal = Journal of Integer Sequences
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| | volume = 9
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| | year = 2006
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| | issue = 4
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| | pages = 06.4.4
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| | id = {{MathSciNet | id = 2247943}}
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| | url = http://www.emis.ams.org/journals/JIS/VOL9/Luca/luca66.pdf}}
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| *{{cite conference
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| | author = Mohan, A. L.; Suryanarayana, D.
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| | title = Perfect totient numbers
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| | booktitle = Number theory (Mysore, 1981)
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| | pages = 101–105
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| | publisher = Lecture Notes in Mathematics, vol. 938, Springer-Verlag
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| | year = 1982
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| | id = {{MathSciNet | id = 0665442}}}}
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| *{{cite journal
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| | author = Venkataraman, T.
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| | title = Perfect totient number
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| | journal = [[The Mathematics Student]]
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| | volume = 43
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| | year = 1975
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| | page = 178
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| | id = {{MathSciNet | id = 0447089}}}}
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| {{PlanetMath attribution|id=8741|title=Perfect Totient Number}}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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Hello and welcome. My name is Numbers Wunder. North Dakota is our beginning location. To play baseball is the hobby he will never quit performing. Supervising is my profession.
My web-site - home std test