Permutation Of Some Primes

[Leroy Quet]

I have posted the message below to sci.math and rec.puzzles.
I post it to seq.fan because of the final question regarding a sequence.

--


[crossposted to rec.puzzles because I leave the 10-prime example I found 
unrevealed for now...]

If you take the first n primes, you can (for some n) find a permutation 
of them such that the greatest prime divisors of each sum of consecutive 
primes (consecutive in regards to their placement in the permutation) 
forms a permutation of the first (n-1) primes 
(so that each greatest prime-divisor q, q <= the (n-1)th prime, occurs 
once exactly).

For example, with n = 4, we have:

primes in their permutation:
2, 7, 3, 5 

sums of consecutive primes:
9, 10, 8

greatest prime divisors:
3, 5, 2


But there is no solution for n=3 because we cannot get a greatest prime 
divisor of 3 for any permutation.

I leave as a challenge, which I did myself in not too much time by 
trial-and-error and by-hand,
find the permutation of the first 10 primes (those primes <= 29) so that 
the greatest prime divisors form a permutation of the first 9 primes
(those primes <= 23).


And more generally, what is the sequence which gives how many such 
permutations like this are there for the first n primes?

thanks,
Leroy Quet