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.
***
