ways a number is a sum of 2 distinct primes

[Emeric Deutsch]

Dear Seqfans,

The least number expressible as a sum of two distinct primes in 
exactly n ways is given by sequence A087747, starting with

5, 16, 24, 36, 48, 60, 78, 84, 90, 114, 144, 120, 168, 180, 234,

for n=1,2,3,...
[5=2+3; 16=3+13=5+11; 24=5+19=7+17=11+13]

Numbers which can be expressed as the sum of two distinct primes in 
exactly two ways are given by A077914:
 	16, 18, 20, 22, 26, 28, 32, 62, 68
Numbers which can be expressed as the sum of two distinct primes in 
exactly three ways are given by A077969:
 	24, 30, 34, 40, 44, 46, 52, 56, 58, 98, 122, 128
Numbers which can be expressed as the sum of two distinct primes in 
exactly four ways are given by A078299:
 	36, 42, 50, 74, 80, 82, 86, 88, 92, 94, 152, 158
and so on
5 ways, A080854: 48, 54, 64, 70, 76, 104, 106, 118, 124, 134, 136, 146, 
148, 164, 166, 188;

6 ways, A080862: 60, 66, 72, 100, 110, 116, 172, 178, 182, 194, 206, 212, 
218, 226, 248, 278, 326, 332, 398

7 ways, not in OEIS: 78, 96, 112, 130, 140, 142, 176, 208, 214, 224, 232, 
272, 362

8 ways, not in OEIS: 84, 102, 108, 138, 154, 160, 184, 190, 200, 202, 242, 
254, 256, 262, 266, 284, 292, 296, 302, 308, 314, 346, 368, 458

These sequences have no other terms up to 7000. Are they finite?
Is there any literature on this?

So far it SEEMS that the largest number expressible as a sum of two
distinct primes in exactly n =2,3,4,... ways is given by

68,128,158,188,398,362,458,...
(last terms in the finite (?) sequences given above).

I'd appreciate any input on this.
Thanks,
Emeric

P.S. I have used the following g.f. for the number of partitions
of n into 2 distinct primes:

 	sum(sum(x^(p(i)+p(j)), i=1..j-1), j=1..infinity),

where p(k) is the k-th prime. See A117929.