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Kvant Problems
Kvant 2019
M2552
M2552
Part of
Kvant 2019
Problems
(1)
Algebra and NT
Source:
2/12/2022
Let
a
1
,
a
2
,
⋯
a_1,a_2, \cdots
a
1
,
a
2
,
⋯
be a sequence of integers that satisfies:
a
1
=
1
a_1=1
a
1
=
1
and
a
n
+
1
=
a
n
+
a
⌊
n
⌋
,
∀
n
≥
1
a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1
a
n
+
1
=
a
n
+
a
⌊
n
⌋
,
∀
n
≥
1
. Prove that for all positive
k
k
k
, there is
m
≥
1
m \geq 1
m
≥
1
such that
k
∣
a
m
k \mid a_m
k
∣
a
m
.
Sequences
algebra
number theory
Kvant