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Montgomery Blair
MBMT Team Rounds
2020.44
2020.44
Part of
MBMT Team Rounds
Problems
(1)
a_n =sum_{d|n} 1/2^{d+ n/d} 2020 MBMT p44
Source:
2/13/2022
Let
a
n
=
∑
d
∣
n
1
2
d
+
n
d
a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}
a
n
=
∑
d
∣
n
2
d
+
d
n
1
. In other words,
a
n
a_n
a
n
is the sum of
1
2
d
+
n
d
\frac{1}{2^{d+ \frac{n}{d}}}
2
d
+
d
n
1
over all divisors
d
d
d
of
n
n
n
. Find
∑
k
=
1
∞
k
a
k
∑
k
=
1
∞
a
k
=
a
1
+
2
a
2
+
3
a
3
+
.
.
.
.
a
1
+
a
2
+
a
3
+
.
.
.
.
\frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....}
∑
k
=
1
∞
a
k
∑
k
=
1
∞
k
a
k
=
a
1
+
a
2
+
a
3
+
....
a
1
+
2
a
2
+
3
a
3
+
....
number theory
MBMT