Of Composites Between 2 Adjacent Primes

[Leroy Quet]

I have submitted the following sequence (which has yet to appear in the 
EIS):

>%S A113709 4,6,8,12,16,18,20,24,30,36,40,42,44,48,54,60,66,68,72,78
>%N A113709 a(n) is the composite between p(n) and p(n+1), where p(n) is 
>the nth prime, which is divisible by (p(n+1)-p(n)).
>%C A113709 Exactly one composite exists between each p(n+1) and p(n)
>which is divisible by (p(n+1)-p(n)), for n >= 2.
>%e A113709 Between the primes 67 and 71 is the composite 68,
>and 68 is divisible by (71-67)=4. So 68 is in the sequence.
>%Y A113709 A113710
>%O A113709 2
>%K A113709 ,more,nonn,

(I have also submitted the sequence of {the nth term above divided by 
(p(n+1)-p(n))}.)

Could someone please calculate/submit the sequence of terms from the 
above sequence (if extended) that are not the composite (in the run of 
composites between adjacent primes) with the most number of divisors?

For example, all terms I give above, except the 68, are also one of the 
composites (between adjacent primes) which have the most number of 
divisors in their run of composites.
 
So the sequence I am wondering about starts:
68,...

thanks,
Leroy Quet