Permutation Of Some Primes

[Leroy Quet]

I wrote in part:

>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
>
>
>...
>
>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


Another sequence idea based on the above:

Let a(1) = 2;

Let a(m+1), m >= 1, be the lowest prime not among the first m terms of 
the sequence where

the highest prime divisor, b(m), of (a(m)+a(m+1)) is not among the first 
(m-1) b's.

So we have 3 sequences:

a(m):       2, 3, 5, 7, 19, 23, 11, 47, 29,...

a(m)+a(m+1): 5, 8, 12, 26, 42, 34, 58, 76,..

b(m):      5, 2, 3, 13, 7, 17, 29, 19 ,..


(if I did not err.)


I wonder most, in regards to these sequences, are {a(m)} and {b(m)} 
permutations of the primes?

thanks,
Leroy Quet