number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

, a perfect totient number is an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

that is equal to the sum of its iterated

totient In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...

s. That is, we apply the

totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...

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. For example, there are six positive integers less than 9 and

relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

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 , so 9 is a perfect totient number. The first few perfect totient numbers are : 3, 9, 15, 27, 39, 81,

111 111 may refer to: *111 (number) *111 BC *AD 111 *111 (emergency telephone number) *111 (Australian TV channel) * Swissair Flight 111 * ''111'' (Her Majesty & the Wolves album) * ''111'' (Željko Joksimović album) *NHS 111 *(111) a Miller index for ...

183 Year 183 ( CLXXXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Victorinus (or, less frequently, year 936 '' Ab ur ...

, 243,

255 __NOTOC__ Year 255 ( CCLV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Valerianus and Gallienus (or, less frequently, year 1008 '' ...

, 327,

363 __NOTOC__ Year 363 ( CCCLXIII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Iulianus and Sallustius (or, less frequently, year ...

, 471,

729 Year 729 ( DCCXXIX) was a common year starting on Saturday of the Julian calendar. The denomination 729 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for nami ...

, 2187, 2199, 3063, 4359, 4375, ... . In symbols, one writes :

\varphi^i(n) = \begin
\varphi(n), &\text i = 1 \\
\varphi(\varphi^(n)), &\text i \geq 2
\end

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_^ \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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

by 3. All powers of 3 are perfect totient numbers, as may be seen by

induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

using the fact that :

\displaystyle\varphi(3^k) = \varphi(2\times 3^k) = 2 \times 3^.

Venkataraman (1975) found another family of perfect totient numbers: if is

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, then 3''p'' 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, ... . 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^''k''''p'' where ''p'' is prime and ''k'' > 3.

References

* * * * * * * {{Classes of natural numbers Integer sequences