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Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2001 Switzerland Team Selection Test
4
4
Part of
2001 Switzerland Team Selection Test
Problems
(1)
all representations of n as a sum of its distinct divisors,
Source: Switzerland - Swiss TST 2001 p4
2/18/2020
For a natural number
n
≥
2
n \ge 2
n
≥
2
, consider all representations of
n
n
n
as a sum of its distinct divisors,
n
=
t
1
+
t
2
+
.
.
.
+
t
k
,
t
i
∣
n
n = t_1 + t_2 + ... + t_k, t_i| n
n
=
t
1
+
t
2
+
...
+
t
k
,
t
i
∣
n
. Two such representations differing only in order of the summands are considered the same (for example,
20
=
10
+
5
+
4
+
1
20 = 10+5+4+1
20
=
10
+
5
+
4
+
1
and
20
=
5
+
1
+
10
+
4
20 = 5+1+10+4
20
=
5
+
1
+
10
+
4
). Let
a
(
n
)
a(n)
a
(
n
)
be the number of different representations of
n
n
n
in this form. Prove or disprove: There exists M such that
a
(
n
)
≤
M
a(n) \le M
a
(
n
)
≤
M
for all
n
≥
2
n \ge 2
n
≥
2
.
number theory
Sum
Divisors