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National and Regional Contests
Bulgaria Contests
Bulgaria MO Regional Round
2024 Bulgaria MO Regional Round
11.3
NT with difference of divisors
NT with difference of divisors
Source: Bulgaria MO Regional round 2024, 11.3
February 13, 2024
number theory
Problem Statement
A positive integer
n
n
n
is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
if
2
∣
τ
(
n
)
2 \mid \tau(n)
2
∣
τ
(
n
)
and if its divisors are
1
=
d
1
<
d
2
<
…
<
d
2
k
−
1
<
d
2
k
=
n
,
1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n,
1
=
d
1
<
d
2
<
…
<
d
2
k
−
1
<
d
2
k
=
n
,
then
d
k
+
1
−
d
k
=
2
d_{k+1}-d_k=2
d
k
+
1
−
d
k
=
2
and
d
k
+
2
−
d
k
−
1
=
65
d_{k+2}-d_{k-1}=65
d
k
+
2
−
d
k
−
1
=
65
. Find the smallest
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
number.
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