Is the next prime number always the next number divisible by the current prime number, except for any numbers previously divisible by primes?
E.g. take prime number $7$, squared is $49$. The next numbers not previously divisible by $2,3,5$ are $53,59,61,67,71,73,77$ -i.e. the next number divisible by $7$ is $11 \times 7$ - the next prime number times the previous one.
Similarly, take $11$: squared $121$. the next numbers not divisible by $2,3,5,7$ are: $127,131,137,139,143$. i.e. $143$ is the next number divisible by $11$, which is $13 \times 11$, $13$ being the next prime in the sequence.
Is this always the case? Can it be that the next prime number in sequence is not neatly divisible by the previous one or has one in between?
Appreciate this may be a silly question, i'm not a mathematician.