A068307 - decompositions of N into sum of 3 primes

[Chuck Seggelin]

Hello Seqfans,

It has been years since I have written to you.  I was looking at positive 
integers expressed as the sum of three primes tonight.  The sequence is 
already in the OEIS as A068307.  Here's the first few terms and description 
as entered:

=========
A068307 - From Goldbach problem: number of decompositions of n into sum of 
three primes. partitions of n into prime parts and number of elements is 3.

0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 4, 2, 3, 2, 5, 2, 5, 3, 5, 
3, 7, 3, 7, 2, 6, 3, 9, 2, 8, 4, 9, 4, 10, 2, 11, 3, 10, 4, 12, 3, 13, 4, 
12, 5, 15, 4, 16, 3, 14, 5, 17, 3, 16, 4, 16, 6, 19, 3, 21, 5, 20, 6, 20, 2, 
22, 5, 21, 6, 22, 5, 28, 5, 24, 7, 25, 4, 29, 5, 27, 8, 29, 5, 33, 4, 29
=========

I'm particularly interested in those numbers with small numbers of 
decompositions into a sum of three primes, and there appear to be no 
sequences devoted to them:

Number of decompositions = 1: 6, 7, 8, 10, 14
Number of decompositions = 2: 9, 11, 12, 13, 16, 18, 20, 22, 30, 34, 40, 70

The tools I have available to test with are fairly crude, but I've tested up 
to 1010 without finding any more terms.  I could test further, but I'm not 
sure these seqences would be of any interest.

You guys are the experts and have a far better understanding of Goldbach's 
conjectures than I, what's your opinion?  Would the above sequences be of 
any interest? For num decompositions=2 it seems that there would be more 
terms, but for num decompositions=1 I'm less sure, maybe not.

-- Chuck Seggelin