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2016 Harvard-MIT Mathematics Tournament
31
31
Part of
2016 Harvard-MIT Mathematics Tournament
Problems
(1)
2016 Guts #31
Source:
12/24/2016
For a positive integer
n
n
n
, denote by
τ
(
n
)
\tau(n)
τ
(
n
)
the number of positive integer divisors of
n
n
n
, and denote by
ϕ
(
n
)
\phi(n)
ϕ
(
n
)
the number of positive integers that are less than or equal to
n
n
n
and relatively prime to
n
n
n
. Call a positive integer
n
n
n
\emph{good} if
φ
(
n
)
+
4
τ
(
n
)
=
n
\varphi (n) + 4\tau (n) = n
φ
(
n
)
+
4
τ
(
n
)
=
n
. For example, the number
44
44
44
is good because
φ
(
44
)
+
4
τ
(
44
)
=
44
\varphi (44) + 4\tau (44)= 44
φ
(
44
)
+
4
τ
(
44
)
=
44
.Find the sum of all good positive integers
n
n
n
.