Composites Between Adjacent Primes

[Leroy Quet]

Between each pair of adjacent primes, p(n) and p(n+1), there is at least 
one composite divisible by the highest prime dividing any composite 
between p(n) and p(n+1).

So, for example, between 23 and 29 are the composites 24, 25, 26, 27, and 
28.
26 is divisible by 13, and there aren't any primes higher than 13 
dividing any of these composites.

My question is:

Is there always only one composite between each pair of adjacent primes, 
p(n) and p(n+1), which is divisible by the highest prime dividing any 
composite between p(n) and p(n+1)?

The sequence where a(n) = (the highest prime dividing the product of the 
composites between p(n) and p(n+1)) is not in the EIS, apparently.
It starts, if I did not error,
2,3,5,3,7,3,7,13,5,17,...

And the composites divisible by these primes form the sequence, whose 
definition must be modified if there are more than one such composite for 
any particular n,
4,6,10,12,14,18,21,26,30,34,...

Neither of these are in the EIS yet.

(I am an too much of a hurry today to add them myself. SOMEONE should add 
these sequences eventually.)

thanks,
Leroy Quet