11. A sequence $U_{1}, U_{2}, U_{3}, \ldots$ 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 $k \times U_{k}$ ?
Solution
270

The sequence is $2,1,3,4,5,6,7,8,11,9,13,10,17,12,19,14,23,15,29, \ldots$ so that $U_{18}=15$ and $U_{19}=29$. These are the first two consecutive terms with a difference greater than 10 .

Therefore $k=18$ and $k \times U_{k}=18 \times 15=270$.