RE First differences are primes

[Tautócrona]

>could someone please extend this (if of interest for the OEIS) :
>S = 1 4 6 25 30 77 84 95 108 125 148 177 208 245 286 329 382 441 502 573 640 713 792 875 
>964 1065 1162 ...
>Definition :
>« Non-primes sequence whose first differences show all primes, once »

I think the related seq: "primes that appear in seq above, in its order" is also of 
interest. From your own example, the first terms would be:

3, 2, 19, 5...

Would every prime appear on it? Which is the most expected deviation from the original 
place for that prime? This is, if a(n) is the n-th term of this seq,
is |a(n)-Pi(a(n)| bounded? How does a(n)/Pi(a(n)) grow?

Regards. Jose Brox