Euler Totient function, a cornerstone of number theory, has attracted extensive study and applications across many disciplines. In this paper, we explore the patterns that the iterations of the Totient function exhibit. This paper first covers the foundational definitions and well-established theorems. Then, we build upon those results to investigate applying the Totient function multiple times, such as ϕ (ϕ (ϕ (n) ) ). Theorems regarding the end behavior of such iterations are presented. Next, we apply an innovative summation approach to the iterations of the Totient function, which is in the form of ϕ (n) +ϕ (ϕ (n) ) +ϕ (ϕ (ϕ (n) ) ) + that could also be expressed as ϕⁱ (n). We prove novel theorems regarding this sum for all powers of Fermat Primes, and we derive an elegant result for powers of three. This paper initiates investigations into the sums of iterated Totient function values.
Li et al. (2025) studied this question.