Let n;SPMgt;2 be a positive integer and let #tex2html_wrap_inline9# denote Euler's totient function. Define #math1##tex2html_wrap_inline11# and #math2##tex2html_wrap_inline13# for all integers #tex2html_wrap_inline15#. Define the arithmetic function S by #math3##tex2html_wrap_inline19#, where #math4##tex2html_wrap_inline21#. We say n is a perfect totient number if S(n)=n. We give a list of known perfect totient numbers, and we give sufficient conditions for the existence of further perfect totient numbers.