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Putnam
1961 Putnam
A4
A4
Part of
1961 Putnam
Problems
(1)
Putnam 1961 A4
Source: Putnam 1961
6/5/2022
Let
Ω
(
n
)
\Omega(n)
Ω
(
n
)
be the number of prime factors of
n
n
n
. Define
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
and
f
(
n
)
=
(
−
1
)
Ω
(
n
)
.
f(n)=(-1)^{\Omega(n)}.
f
(
n
)
=
(
−
1
)
Ω
(
n
)
.
Furthermore, let
F
(
n
)
=
∑
d
∣
n
f
(
d
)
.
F(n)=\sum_{d|n} f(d).
F
(
n
)
=
d
∣
n
∑
f
(
d
)
.
Prove that
F
(
n
)
=
0
,
1
F(n)=0,1
F
(
n
)
=
0
,
1
for all positive integers
n
n
n
. For which integers
n
n
n
is
F
(
n
)
=
1
?
F(n)=1?
F
(
n
)
=
1
?
Putnam
Arithmetic Functions
number theory