Problems & Puzzles: Puzzles

Puzzle 350. Primes & primorials

Time ago I sent the following curio about the prime 313:

"313 is prime and also the following are primes 313-2, 313-2*3, 313-2*3*5 & 313-2*3*5*7"

This week I ask:

For each q=>2, find the smallest prime p such that p>q# & (p-r#) is prime for all prime r<=q.

This is what I have found in a fast & simple search, up to q=23:

 q p 2 5 3 13 5 43 7 229 11 3463 13 43789 17 1088449 19 19800379 23 264333259 29 9348884059 FF 31 228178314439 JT 37 7931712374479, FS 41 307867708410673, JKA 43 13230211614496609, JKA 47 618681508598750923, JKA

Question: can you extend the table above?

Contributions came from Farideh Firoozbakht, Jacques Tramu, Fred Schneider & J.K.Andersen (see red lines in the table above)

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BTW, this sequence has been posed in OEIS as A115785, by Rick L. Shepherd on Jan, 31, 2006 up to q=23. According to the entries of this sequence, Don Reble computed the p value for q=29 on Feb. 15... Very recent work ignored by me last week!!!

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