perfect totient number

{{Short description|Number that is the sum of its iterated totients}}

In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

Examples

For example, there are six positive integers less than 9 and relatively prime to it, so the totient of 9 is 6; there are two numbers less than 6 and relatively prime to it, so the totient of 6 is 2; and there is one number less than 2 and relatively prime to it, so the totient of 2 is 1; and {{math|9 {{=}} 6 + 2 + 1}}, so 9 is a perfect totient number.

The first few perfect totient numbers are

:3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, ... {{OEIS|id=A082897}}.

Notation

In symbols, one writes

: $\varphi^i(n) = \begin{cases}
\varphi(n), &\text{ if } i = 1 \\
\varphi(\varphi^{i-1}(n)), &\text{ if } i \geq 2
\end{cases}$

for the iterated totient function. Then if c is the integer such that

: $\displaystyle\varphi^c(n)=2,$

one has that n is a perfect totient number if

: $n = \sum_{i = 1}^{c + 1} \varphi^i(n).$

Multiples and powers of three

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

: $\displaystyle\varphi(3^k) = \varphi(2\times 3^k) = 2 \times 3^{k-1}.$

Venkataraman (1975) found another family of perfect totient numbers: if {{math|p {{=}} 4 × 3^k + 1}} is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are

:0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... {{OEIS|id=A005537}}.

More generally if p is a prime number greater than 3, and 3p 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 9p 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^kp where p is prime and k > 3.

References

{{cite journal

| author = Pérez-Cacho Villaverde, Laureano

| title = Sobre la suma de indicadores de ordenes sucesivos

| journal = Revista Matematica Hispano-Americana

| volume = 5

| issue = 3

| year = 1939

| pages = 45–50}}

{{cite book

| author = Guy, Richard K.

| author-link = Richard K. Guy

| title = Unsolved Problems in Number Theory

| location = New York

| publisher = Springer-Verlag

| year = 2004

| isbn=0-387-20860-7

| page = §B41}}

{{cite journal

| author1 = Iannucci, Douglas E.

| author2 = Deng, Moujie

| author3 = Cohen, Graeme L.

| title = On perfect totient numbers

| journal = Journal of Integer Sequences

| volume = 6

| year = 2003

| issue = 4

| pages = 03.4.5

| bibcode = 2003JIntS...6...45I

| mr = 2051959

| url = http://www.emis.de/journals/JIS/VOL6/Cohen2/cohen50.pdf

| access-date = 2007-02-07

| archive-date = 2017-08-12

| archive-url = https://web.archive.org/web/20170812121811/http://www.emis.de/journals/JIS/VOL6/Cohen2/cohen50.pdf

| url-status = dead

}}

{{cite journal

| author = Luca, Florian

| author-link = Florian Luca

| title = On the distribution of perfect totients

| journal = Journal of Integer Sequences

| volume = 9

| year = 2006

| issue = 4

| pages = 06.4.4

| bibcode = 2006JIntS...9...44L

| mr = 2247943

| url = http://www.emis.de/journals/JIS/VOL9/Luca/luca66.pdf

| access-date = 2007-02-07

}}

{{cite conference

|author1=Mohan, A. L. |author2=Suryanarayana, D. | title = Perfect totient numbers

| book-title = Number theory (Mysore, 1981)

| pages = 101–105

| publisher = Lecture Notes in Mathematics, vol. 938, Springer-Verlag

| year = 1982

|mr=0665442}}

{{cite journal

| author = Venkataraman, T.

| title = Perfect totient number

| journal = The Mathematics Student

| volume = 43

| year = 1975

| page = 178

|mr=0447089}}

{{cite web

| url = https://trepo.tuni.fi/handle/10024/97744

| title = Täydelliset totienttiluvut

| author = Hyvärinen, Tuukka

| date = 2015

| publisher = Tampereen yliopisto

| location = Tampere}}

Category:Integer sequences