1,1

It is trivial that perfect totient numbers must be odd. It is easy to show that powers of 3 are perfect totient numbers.

L. Perez Cacho, "Sobre la suma de indicadores de ordenes sucesivos", Revista Matematica Hispano-Americana, 5.3 (1939), 45-50.

A. L. Mohan and D. Suryanarayana, "Perfect totient numbers", in: Number Theory (Proc. Third Matscience Conf., Mysore, 1981) Lecture Notes in Math. 938 (Springer-Verlag, New York, 1982) pp. 101-105.

Igor E. Shparlinski, On the Sum of Iterations of the Euler Function, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.6.

Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, On Perfect Totient Numbers, J. Integer Sequences, 6 (2003), #03.4.5.

n is a perfect totient number if S(n)=n, where S(n)=phi(n)+phi^2(n)+ . . . +1, where phi is Euler's totient function, and phi^2(n)=phi(phi(n)), . . ., phi^k(n)=phi(phi^(k-1)(n)).

n such that n = A092693(n)

n such that 2*n = A053478(n). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 02 2004

327 is a perfect totient number because

327=216+72+24+8+4+2+1. Note that 216=phi(327), 72=phi(216), 24=phi(72), and so on.

kMax=57395631; a=Table[0, {kMax}]; lst={}; Do[e=EulerPhi[k]; a[[k]]=e+a[[e]]; If[k==a[[k]], AppendTo[lst, k]], {k, 2, kMax}]; lst

Cf. A092693 (sum of iterated phi(n)). See also A091847.

Adjacent sequences: A082894 A082895 A082896 this_sequence A082898 A082899 A082900

Sequence in context: A055927 A087031 A089632 this_sequence A131822 A131801 A122819

nonn

Douglas E. Iannucci (diannuc(AT)uvi.edu), Jul 21 2003

Corrected by T. D. Noe (noe(AT)sspectra.com), Mar 11 2004