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

[Richard Guy]

I believe that your conjectures follow from a
more general one by Hardy & Littlewood (probably
in Some problems of `partitio numerorum' III,
on the expression of a number as a sum of primes,
Acta Math 44(1922) 1--70).

They give a heuristic for the number of
representations, involving the log function,
so less than k for only finitely many values.

R.

On Mon, 14 Jan 2002, Peter Bertok wrote:

> [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
> 
> 
> 
>