# Finding the Integer

Number Theory Level 5

For any integer $$k\ge1$$, let $$p(k)$$ be the smallest prime which does not divide $$k$$. Define the integer function $$X(k)$$ to be the product of all primes less than $$p(k)$$ if $$p(k)>2$$, and $$X(k)=1$$ if $$p(k)=2$$. Let $$\{x_n\}$$ be the sequence defined by $$x_0=1$$, and $$x_{n+1}X(x_n)=x_np(x_n)$$ for $$n\ge0$$. Find the smallest positive integer, $$t$$ such that $$x_t=2090$$.

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