On the new sequences A067187, A067188, A067189, A067190, and A067191

[N. J. A. Sloane]

Peter Bertok said:


    All of these sequence are about "integers expressible as the sum of 2
primes in 'n' different ways", where n is some small number. Eg:

A067189 (n = 1)
    4, 6, 7, 8, 9, 12, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61,
63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141,
151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229,
231, 235, 241, 243, 253, 259, 265, 271, 273, 279, 283, 285, 295, 309, 313,
315, 319, 333, 339, 349, 351, 355, 361, 369, 375, 381, 385, 391, 399, 403,
411, 421, 423, 433, 435, 441, 445, 451, 459, ...

Note: For the following (and all higher values of 'n'), I propose the
conjecture that the sequences are finite:

A067189 (n = 2)
    5, 10, 14, 16, 18, 20, 28, 32, 38, 68

A067189 (n = 3)
    22, 24, 26, 30, 40, 44, 52, 56, 62, 98, 128

A067190 (n = 4)
    34, 36, 42, 46, 50, 58, 80, 88, 92, 122, 152

A067191 (n = 5)
    48, 54, 64, 70, 74, 76, 82, 86, 94, 104, 124, 136, 148, 158, 164, 188

...

However, that can't be right.
The original version of A067187 (note correction
to A number) was
%S A067187 4,5,6,7,8,9,12,13,15,19,21,25,31,43,61,73,103,109,139,181,193,199,229,
%T A067187 241,271,283,313,349,421

and i like that better

after all, 5 = 2+3 in just one way

Peter is counting ordered sums

I will add his as separte seqs.

NJAS