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Silk Road
2018 Silk Road
4
4
Part of
2018 Silk Road
Problems
(1)
Exactly once sequence
Source: SRMO 2018 P4
8/22/2018
Does there exist a sequence of positive integers
a
1
,
a
2
,
.
.
.
a_1,a_2,...
a
1
,
a
2
,
...
such that every positive integer occurs exactly once and that the number
τ
(
n
a
n
+
1
n
+
(
n
+
1
)
a
n
n
+
1
)
\tau (na_{n+1}^n+(n+1)a_n^{n+1})
τ
(
n
a
n
+
1
n
+
(
n
+
1
)
a
n
n
+
1
)
is divisible by
n
n
n
for all positive integer.Here
τ
(
n
)
\tau (n)
τ
(
n
)
denotes the number of positive divisor of
n
n
n
.
number theory