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ON THE ITERATES OF THE SHIFTED EULER’S FUNCTION
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 12 May 2023, pp. 206-214
- Print publication:
- April 2024
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- Article
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- You have access
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Let
$\varphi $ be Euler’s function and fix an integer 
$k\ge 0$. We show that for every initial value 
$x_1\ge 1$, the sequence of positive integers 
$(x_n)_{n\ge 1}$ defined by 
$x_{n+1}=\varphi (x_n)+k$ for all 
$n\ge 1$ is eventually periodic. Similarly, for all initial values 
$x_1,x_2\ge 1$, the sequence of positive integers 
$(x_n)_{n\ge 1}$ defined by 
$x_{n+2}=\varphi (x_{n+1})+\varphi (x_n)+k$ for all 
$n\ge 1$ is eventually periodic, provided that k is even.
