Suppose that the sequence {an} of positive integers satisfies the following conditions: [*]For an integer i≥2022, define ai as the smallest positive integer x such that x+∑k=i−2021i−1ak is a perfect square.
[*]There exists infinitely many positive integers n such that an=4×2022−3. Prove that there exists a positive integer N such that ∑k=nn+2021ak is constant for every integer n≥N.
And determine the value of ∑k=NN+2021ak. number theoryPerfect SquaresSequenceInteger sequence