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Let

d(n)

denote the number of positive divisors of

n

. For any given integer

a \geq 3

, define a sequence

\{a_i\}_{i=0}^\infty

satisfying
[*]

a_{0}=a

, and [*]

a_{n+1}=a_{n}+(-1)^{n} d(a_{n})

for each integer

n \geq 0

.
For example, if

a=275

, the sequence would be

275, \overline{281,279,285,277,279,273}.

Prove that for each positive integer

k

there exists a positive integer

N

such that if such a sequence has period

2k

and all terms of the sequence are greater than

N

then all terms of the sequence have the same parity.
Proposed by Navid

2