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2015 BMT Spring
12
2015 BMT Team 12
2015 BMT Team 12
Source:
January 6, 2022
number theory
Problem Statement
Let
f
(
n
)
f(n)
f
(
n
)
be the number of ordered pairs
(
k
,
ℓ
)
(k, \ell)
(
k
,
ℓ
)
of positive integers such that
n
=
(
2
ℓ
−
1
)
⋅
2
k
−
k
n = (2\ell-1)\cdot 2^k - k
n
=
(
2
ℓ
−
1
)
⋅
2
k
−
k
, and let
g
(
n
)
g(n)
g
(
n
)
be the number of ordered pairs
(
k
,
ℓ
)
(k, \ell)
(
k
,
ℓ
)
of positive integers such that
n
=
ℓ
⋅
2
k
+
1
−
k
n = \ell \cdot 2^{k+1}-k
n
=
ℓ
⋅
2
k
+
1
−
k
. Compute
∑
i
=
1
∞
f
(
i
)
−
g
(
i
)
2
i
\sum_{i=1}^{\infty}\frac{f(i) - g(i)}{2^i}
∑
i
=
1
∞
2
i
f
(
i
)
−
g
(
i
)
..
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