Senior Kangaroo 2018 Q11

A sequence $U_1$, $U_2$, $U_3$, ... is defined as follows:

• $U_1=2$;• if $U_n$ is prime then $U_{n+1}$ is the smallest positive integer not yet in the sequence;• if $U_n$ is not prime then $U_{n+1}$ is the smallest prime not yet in the sequence.

The integer $k$ is the smallest such that $U_{k+1}-U_k>10$.

What is the value of $k\times U_k$?