examples of perfect totient numbers

As has been proven, all powers of 3 are perfect totient numbers. Thus 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, etc. are all examples of perfect totient numbers.

Digging a little deeper, it follows from Perez Cacho’s proof that $3p$ is a perfect totient number when the prime is of the form $4n+1$, where $n$ is a smaller perfect totient number, that perfect totient numbers can be chained together in a somewhat similar manner to Sophie Germain primes and safe primes in Cunningham chains. That is, if $n$ is a perfect totient number and $4n+1$ is a prime, then $12n+3$ is a perfect totient number. Although 1 is technically not a perfect totient number, since it is technically an integer power of 3, if we indulge in granting it perfect totient number status for the purpose of seeking chains, we find that it generates the chain 1, 15, 183, 2199.

The actual powers of 3 that are in fact perfect totient numbers disappoint when it comes to generating chains: 3, 39, 471; 9, 111; 27, 327; we need not even consider those $3^x\equiv 1mod5$ because in that case then clearly $5|(4\cdot 3^x+1)$; 243; 729, 8751; etc.

We’d be led astray, however, if we thought that we could generate a complete listing of perfect totient numbers simply by trying to make chains from powers of 3; such a method would overlook 4375 (which alas, however, does not make a chain). Neither does 4190263, 3932935775 nor 4764161215, three other perfect totient numbers that are not divisible by 3 and which do not give primes when multiplied by 4 and added 1.

How about other multiples of 3? There is 255, 3063, 36759; 363, 4359; if nothing else among chain beginnings below a million.