More on A067187, A067188, A067189, A067190, and A067191 [Corrected]

[Brian L. Galebach]

Sequence A056636 is not quite the same as A000954 since A000954 is for
"exactly n ways", while A056636 is for "at most n ways".  However, one could
rightly argue that A000954 is a more useful sequence since it contains more
information.

Brian L. Galebach

-----Original Message-----
From: Peter Bertok [mailto:peter at bertok.com]
Sent: Sunday, January 13, 2002 6:26 PM
To: seqfan at ext.jussieu.fr
Subject: Re: More on A067187, A067188, A067189, A067190, and A067191
[Corrected]


[This post has had corrections made, thanks go to NJS for finding my error!
Corrections in square brackets]

    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, 5, 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, ...

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

A067189 (n = 2)
    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

A067191 (n = 6)
    60, 66, 72, 100, 106, 110, 116, 118, 134, 146, 166, 172, 182, 212, 248,
332

[
The first term of each of those sequences (4,10,22,34,48,60) is A067184.

For the following, consider only the finite sequences:

The last term (68, 128, 152, 188, 332) in each sequence is the sequence
A000954 starting from the third position. (and A056636, which is probably
redundant).

While we're looking for other relationships, the length of each sequence (9,
11, 11, 16, 16) is A000974 starting from the third offset.

(Note: "even integers" doesn't mean much in the cases with n > 2, since
there is only one even prime, and the sum of two odd primes must be even.
Hence, all the terms of the sequences with n > 2 must be even.)
]

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 its 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