Problems & Puzzles: Puzzles

Puzzle 181. (N+k)/k primes

I received from Phil Carmody the following nice puzzle:

Find the least N such that (N+k)/k is prime for k=1, 2, ..., n, for a given value n

The already known results are:

 n N Author 1 1 Jack Brennen 2 2 "   " 3 12 "   " 4 12720 "   " 5 19440 "   " 6 5516280 "   " 7 5516280 "   " 8 7321991040 "   " 9 363500177040 "   " 10 2394196081200 "   " 11 3163427380990800 "   " 12 22755817971366480 Phil Carmody 13 3788978012188649280 "   " 14 2918756139031688155200 J. K. Andersen (20/4/03)

Example:

for n=3, (12+1)/1, (12+2)/2 & (12+3)/3 are primes while (12+4)/4 is composite

Phil warns me that:

"Jack Brennen found terms up to a=11, and I found the a=12 and a=13 term, but Jud claims he recognizes it from somewhere else - have you seen it before? My length 13 result was only 30 minutes work, so maybe if it's not an old puzzle, it could be turned into a new one!"

Question:

1) Do you know if this sequence has been studied and published before and where?
2) Can you find the next 3 members of the sequence?

Solution:

J. K. Andersen found (20/4/03) the solution for n=14:

I agree with the given results and add:
The smallest solution for n=14 is N=2918756139031688155200 (22 digits).
Found in 8 days with the C program written for puzzle 206.

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