Question 15
For any integer $k \ge 1,$ 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 $p(k) \gt 2,$ and $X(k) = 1$ if $p(k) = 2.$ Let $\{x_n\}$ be the sequence defined by $x_0 = 1,$ and $x_{n+1} X(x_n) = x_n p(x_n)$ for $n \ge 0.$ Find the smallest positive integer $t$ such that $x_t = 2090.$