Perfect totient numbers

Generate and show here, the first twenty Perfect totient numbers.

Perfect totient numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Python

<lang python>from math import gcd from functools import lru_cache from itertools import islice, count

@lru_cache(maxsize=None) def φ(n):

   return sum(1 for k in range(1, n + 1) if gcd(n, k) == 1)

def perfect_totient():

   for n0 in count(1):
       parts, n = 0, n0
       while n != 1:
           n = φ(n)
           parts += n
       if parts == n0:
           yield n0
       

if __name__ == '__main__':

   print(list(islice(perfect_totient(), 20)))</lang>

Output:

[3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]