Innumber theory, aperfect totient number is aninteger that is equal to the sum of its iteratedtotients. That is, one applies thetotient function to a numbern, 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 equalsn, thenn is a perfect totient number.

For example, there are six positive integers less than 9 andrelatively 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; and9 = 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, ... (sequenceA082897 in theOEIS).

In symbols, one writes

for the iterated totient function. Then ifc is the integer such that

${\displaystyle {\displaystyle {\phi }^c(n)=2,}}$

one has thatn is a perfect totient number if

${\displaystyle n=\sum _{i=1}^{c+1}{\phi }^i(n).}$

It can be observed that many perfect totient aremultiples of 3; in fact, 4375 is the smallest perfect totient number that is notdivisible by 3. Allpowers of 3 are perfect totient numbers, as may be seen byinduction using the fact that

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

Venkataraman (1975) found another family of perfect totient numbers: ifp = 4 × 3^k + 1 isprime, then 3p is a perfect totient number. The values ofk leading to perfect totient numbers in this way are

0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... (sequenceA005537 in theOEIS).

More generally ifp is a prime number greater than 3, and 3p is a perfect totient number, thenp ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not allp 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 thenp 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 wherep is prime andk > 3.

• Pérez-Cacho Villaverde, Laureano (1939). "Sobre la suma de indicadores de ordenes sucesivos".Revista Matematica Hispano-Americana.5 (3):45–50.

• Guy, Richard K. (2004).Unsolved Problems in Number Theory. New York: Springer-Verlag. p. §B41.ISBN 0-387-20860-7.

• Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L. (2003)."On perfect totient numbers"(PDF).Journal of Integer Sequences.6 (4): 03.4.5.Bibcode:2003JIntS...6...45I.MR 2051959. Archived fromthe original(PDF) on 2017-08-12. Retrieved2007-02-07.

• Luca, Florian (2006)."On the distribution of perfect totients"(PDF).Journal of Integer Sequences.9 (4): 06.4.4.Bibcode:2006JIntS...9...44L.MR 2247943. Retrieved2007-02-07.

• Mohan, A. L.; Suryanarayana, D. (1982). "Perfect totient numbers".Number theory (Mysore, 1981). Lecture Notes in Mathematics, vol. 938, Springer-Verlag. pp. 101–105.MR 0665442.

• Venkataraman, T. (1975). "Perfect totient number".The Mathematics Student.43: 178.MR 0447089.

• Hyvärinen, Tuukka (2015)."Täydelliset totienttiluvut". Tampere: Tampereen yliopisto.

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