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

Peter Bertok peter at bertok.com
Sun Jan 13 08:16:06 CET 2002

```    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

I can't prove the conjecture, but brute-force testing up to 10,000 can't
find any more terms, and a simply taking a look at the graph of the 'n',
it's obvious that it's value is always either 0, 1 or k1*ln(n) to k2*ln(n),
which implies that after a while, there will be no further solutions for any
finite value of n. The first 1000 values are:

0, 0, 1, 2, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 3, 0, 3, 1, 3,
0, 2, 0, 3, 1, 2, 1, 4, 0, 4, 0, 2, 1, 3, 0, 4, 1, 3, 1, 4, 0, 5, 1, 4, 0,
3, 0, 5, 1, 3, 0, 4, 0, 6, 1, 3, 1, 5, 0, 6, 0, 2, 1, 5, 0, 6, 1, 5, 1, 5,
0, 7, 0, 4, 1, 5, 0, 8, 1, 5, 0, 4, 0, 9, 1, 4, 0, 5, 0, 7, 0, 3, 1, 6, 0,
8, 1, 5, 1, 6, 0, 8, 1, 6, 1, 7, 0, 10, 1, 6, 0, 6, 0, 12, 0, 4, 0, 5, 0,
10, 0, 3, 1, 7, 0, 9, 1, 6, 0, 5, 0, 8, 1, 7, 1, 8, 0, 11, 0, 6, 0, 5, 0,
12, 1, 4, 1, 8, 0, 11, 0, 5, 1, 8, 0, 10, 0, 5, 1, 6, 0, 13, 1, 9, 0, 6, 0,
11, 1, 7, 0, 7, 0, 14, 1, 6, 1, 8, 0, 13, 0, 5, 0, 8, 0, 11, 1, 7, 1, 9, 0,
13, 1, 8, 1, 9, 0, 14, 0, 7, 0, 7, 0, 19, 0, 6, 1, 8, 0, 13, 0, 7, 0, 9, 0,
11, 0, 7, 1, 7, 0, 12, 1, 9, 1, 7, 0, 15, 1, 9, 0, 9, 0, 18, 1, 8, 1, 9, 0,
16, 0, 6, 0, 9, 0, 16, 1, 9, 0, 8, 0, 14, 1, 10, 0, 9, 0, 16, 1, 8, 0, 9, 0,
19, 1, 7, 1, 11, 0, 16, 0, 7, 1, 14, 0, 16, 1, 8, 1, 12, 0, 17, 0, 10, 0, 8,
0, 19, 1, 8, 0, 11, 0, 21, 0, 9, 0, 10, 0, 15, 0, 8, 1, 12, 0, 17, 1, 9, 1,
10, 0, 15, 1, 11, 0, 11, 0, 20, 0, 7, 0, 10, 0, 24, 0, 6, 1, 11, 0, 19, 0,
9, 1, 13, 0, 17, 0, 10, 0, 9, 0, 16, 1, 13, 1, 10, 0, 20, 1, 9, 0, 10, 0,
22, 1, 8, 0, 14, 0, 18, 0, 8, 1, 14, 0, 18, 0, 10, 1, 11, 0, 22, 0, 13, 1,
10, 0, 19, 1, 12, 0, 9, 0, 27, 1, 11, 0, 11, 0, 21, 0, 7, 1, 14, 0, 17, 1,
11, 0, 13, 0, 20, 0, 13, 1, 11, 0, 21, 0, 10, 0, 11, 0, 30, 1, 11, 1, 12, 0,
21, 0, 9, 0, 14, 0, 19, 1, 13, 1, 11, 0, 21, 0, 14, 1, 13, 0, 21, 1, 12, 0,
13, 0, 27, 1, 12, 0, 12, 0, 24, 0, 9, 1, 16, 0, 28, 1, 12, 1, 13, 0, 24, 1,
15, 0, 13, 0, 23, 0, 14, 0, 11, 0, 29, 1, 11, 0, 14, 0, 23, 0, 9, 1, 19, 0,
22, 1, 13, 0, 13, 0, 23, 0, 13, 1, 15, 0, 27, 1, 15, 0, 14, 0, 32, 1, 11, 0,
14, 0, 23, 0, 11, 0, 17, 0, 24, 1, 11, 1, 15, 0, 25, 0, 14, 0, 17, 0, 22, 0,
13, 0, 14, 0, 30, 0, 10, 1, 13, 0, 30, 0, 11, 1, 19, 0, 23, 0, 11, 0, 11, 0,
23, 1, 18, 0, 14, 0, 24, 1, 13, 0, 13, 0, 31, 1, 11, 1, 16, 0, 26, 0, 12, 1,
19, 0, 25, 0, 12, 0, 13, 0, 29, 1, 16, 0, 15, 0, 27, 1, 12, 0, 15, 0, 32, 1,
12, 1, 14, 0, 27, 0, 13, 1, 20, 0, 26, 0, 15, 1, 19, 0, 26, 1, 18, 1, 17, 0,
31, 0, 12, 0, 16, 0, 41, 0, 10, 1, 14, 0, 28, 0, 15, 0, 18, 0, 25, 1, 17, 1,
16, 0, 27, 1, 21, 0, 15, 0, 29, 1, 13, 0, 19, 0, 41, 1, 14, 1, 16, 0, 31, 0,
11, 0, 21, 0, 33, 0, 15, 1, 17, 0, 28, 1, 21, 0, 16, 0, 30, 1, 16, 0, 16, 0,
39, 0, 11, 1, 19, 0, 30, 0, 14, 0, 24, 0, 31, 1, 18, 0, 19, 0, 24, 0, 16, 1,
17, 0, 37, 0, 14, 0, 15, 0, 39, 1, 14, 0, 15, 0, 31, 0, 15, 1, 21, 0, 31, 0,
15, 1, 19, 0, 29, 0, 18, 1, 19, 0, 31, 1, 18, 0, 19, 0, 39, 0, 14, 1, 17, 0,
35, 0, 15, 1, 21, 0, 30, 1, 17, 0, 17, 0, 31, 0, 26, 1, 18, 0, 32, 1, 16, 0,
15, 0, 44, 0, 14, 0, 18, 0, 30, 0, 15, 1, 22, 0, 34, 0, 17, 0, 14, 0, 38, 1,
21, 0, 16, 0, 32, 0, 16, 0, 14, 0, 39, 1, 18, 1, 20, 0, 34, 0, 17, 0, 20, 0,
29, 1, 16, 1, 21, 0, 34, 1, 22, 1, 22, 0, 33, 0, 18, 0, 17, 0, 51, 1, 18, 0,
17, 0, 32, 0, 15, 0, 25, 0, 31, 0, 20, 1, 19, 0, 39, 1, 18, 1, 17, 0, 33, 1,
17, 0, 21, 0, 46, 0, 18, 0, 19, 0, 36, 0, 14, 1, 25, 0, 39, 1, 21, 1, 18, 0,
37, 1, 23, 0, 19, 0, 34, 0, 20, 0, 19, 0, 48, 0, 15, 0, 17, 0, 34, 0, 15, 1,
31, 0, 31, 1, 20, 0, 18, 0, 35, 0, 23, 1, 20, 0, 47, 0, 18, 0, 18, 0, 43, 1,
17, 0, 20, 0, 36, 0, 18, 1, 24, 0, 34, 1, 18, 0, 20, 0, 33, 1, 25, 0, 23, 0,
37, 1, 19, 0, 22, 0, 45, 0, 16, 0, 18, 0, 45, 0, 17, 1, 27, 0, 32, 1, 17, 0,
19, 0, 35, 1, 26, 0, 17, 0, 39, 1, 20, 0, 23, 0, 52, 0, 13, 1, 25, 0, 37, 0,
17, 1, 28

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