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ITAMO
2005 ITAMO
2
For which h, a_n=1
For which h, a_n=1
Source: ItaMO 2005, P5
March 9, 2012
modular arithmetic
Euler
function
number theory
totient function
algebra proposed
algebra
Problem Statement
Let
h
h
h
be a positive integer. The sequence
a
n
a_n
a
n
is defined by
a
0
=
1
a_0 = 1
a
0
=
1
and
a
n
+
1
=
{
a
n
2
if
a
n
is even
a
n
+
h
otherwise
.
a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}
a
n
+
1
=
{
2
a
n
if
a
n
is even
a
n
+
h
otherwise
.
For example,
h
=
27
h = 27
h
=
27
yields
a
1
=
28
,
a
2
=
14
,
a
3
=
7
,
a
4
=
34
a_1=28, a_2 = 14, a_3 = 7, a_4 = 34
a
1
=
28
,
a
2
=
14
,
a
3
=
7
,
a
4
=
34
etc. For which
h
h
h
is there an
n
>
0
n > 0
n
>
0
with
a
n
=
1
a_n = 1
a
n
=
1
?
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