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2013 Harvard-MIT Mathematics Tournament
18
18
Part of
2013 Harvard-MIT Mathematics Tournament
Problems
(1)
2013 HMMT Guts #18: Multi-Base Recursion
Source:
3/26/2013
Define the sequence of positive integers
{
a
n
}
\{a_n\}
{
a
n
}
as follows. Let
a
1
=
1
a_1=1
a
1
=
1
,
a
2
=
3
a_2=3
a
2
=
3
, and for each
n
>
2
n>2
n
>
2
, let
a
n
a_n
a
n
be the result of expressing
a
n
−
1
a_{n-1}
a
n
−
1
in base
n
−
1
n-1
n
−
1
, then reading the resulting numeral in base
n
n
n
, then adding
2
2
2
(in base
n
n
n
). For example,
a
2
=
3
10
=
1
1
2
a_2=3_{10}=11_2
a
2
=
3
10
=
1
1
2
, so
a
3
=
1
1
3
+
2
3
=
6
10
a_3=11_3+2_3=6_{10}
a
3
=
1
1
3
+
2
3
=
6
10
. Express
a
2013
a_{2013}
a
2013
in base
10
10
10
.
HMMT