Journal of Integer Sequences, Vol. 6 (2003), Article 03.4.5

On Perfect Totient Numbers

Douglas E. Iannucci
University of the Virgin Islands
St Thomas, VI 00802

Deng Moujie
Science and Engineering College
Hainan University
Haikou City 570228
P. R. China

Graeme L. Cohen
University of Technology, Sydney
PO Box 123, Broadway
NSW 2007


Let n>2 be a positive integer and let $\phi$ denote Euler's totient function. Define $\phi^1(n)=\phi(n)$ and $\phi^k(n)=\phi(\phi^{k-1}(n))$ for all integers $k\ge2$. Define the arithmetic function S by $S(n)=\phi(n)+\phi^2(n)+\cdots+\phi^c(n)+1$, where $\phi^c(n)=2$. 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.

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Received July 23 2003; revised version received October 2 2003. Published in Journal of Integer Sequences December 18 2003. Minor typo corrected, June 8 2009.

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