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ON THE ITERATES OF THE SHIFTED EULER’S FUNCTION
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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 $(x_n)_{n\ge 1}$… Click to show full abstract
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.
