Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.6

On the Sum of Iterations of the Euler Function

Igor E. Shparlinski
Department of Computing
Macquarie University
Sydney, NSW 2109


We study the sum

F(n) = \sum_{k=1}^{\kappa(n)} \varphi^{(k)}(n) .

of consecutive iterations of the Euler function $\varphi(n)$ (where the last iteration satisfies $\varphi^{(\kappa(n))}(n)=1$). We show that for almost all $n$, the difference $\vert F(n) - n\vert$ is not too small, and the ratio $n/F(n)$ is not an integer. The latter result is related to a question about the so-called perfect totient numbers, for which $F(n) = n$.

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(Concerned with sequences A082897 .)

Received July 27 2005; revised version received January 18 2006. Published in Journal of Integer Sequences January 23 2006.

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