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.