Sloane's A077463 obviously suggests that for any positive integer $n$ there exist $n$ consecutive primes and only them in between $m$ and $2m$ for some natural number $m$.

For instance, for

$n=1$, take $m=2$; $\hspace{.2in}$$2<3<4$;

$n=2$, take $m=7$; $\hspace{.2in}$$7<11,13<14$;

$n=3$, take $m=9$; $\hspace{.2in}$$9<11,13,17<18$;

$n=4$, take $m=15$; $\hspace{.1in}$$15<17,19,23,29<30$,$\hspace{.1in}$ etc.

This problem offers a partial converse(given the number of primes, one seeks an exact interval $[m,2m]$) to Bertrand's Postulate(given an interval, one seeks at least one prime in it).

I would like to know

whether this problem is solved, or

whether there are stronger known conjectures to which it is a consequence.

Thanks, as always