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Prove that there exists an infinite sequence of positive integers

a_1, a_2, ...

such that for all positive integers

i

, \\ i)

a_{i + 1}

is divisible by

a_{i}

.\\ ii)

a_i

is not divisible by

3

.\\ iii)

a_i

is divisible by

2^{i + 2}

but not

2^{i + 3}

.\\ iv)

6a_i + 1

is a prime power.\\ v)

a_i

can be written as the sum of the two perfect squares.

Problem Statement