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?


Of course, we can ask about such permutations, finite or infinite, where 
the highest-prime-divisor sequence is also such a permutation.


Does anyone have a significant finite-lengthed example?


We can construct infinite sequences which MAY be permutations of the 
primes as follows:

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, and...

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

a(m):         2, 3, 5, 7, 19, 31,...

a(m)+a(m+1):  5, 8, 12, 26, 50,...

b(m):         5, 2, 3, 13, 5,...

b(m)+b(m+1):  7, 5, 16, 18,...

c(m):         7, 5, 2, 3,...

(I believe. figured by hand.)

And we can continue downward with an a-sequence depending on more and 
more sequences of primes necessarily distinct.

thanks,
Leroy Quet