Sequence related to Goldbach Conjecture

[Andrew Plewe]

Actually, on further examination your post does lead to what, I think, is a more efficient method for producing the even integers
using only odd integers (including nonprimes).  If you take the rows and columns of your table and make them into a single sequence:

0, 1, 2, 4, 6, 7, 12, 14, 16, 24, etc.

Then subtract 1 from each of the even terms and eliminate the first three terms of the new sequence (-1, 0, 1):

2, 3, 5, 7, 11, 13, 15, 23, etc. (I've inserted 2 just for completeness, to cover all even integers >= 4)

This seems to produce the desired table.  I've verified up to the first thirty terms that it covers all the even numbers.  When I
have time later I'll look at the sequences it's based on to determine if there's a proof.  I'll also submit the new sequence to the
OEIS (assuming it holds up).  Thanks for your help!

	-Andrew Plewe-


-----Original Message-----
From: Don Reble [mailto:djr at nk.ca] 
Sent: Wednesday, March 30, 2005 10:48 AM
To: Andrew Plewe
Cc: 'Seqfan'
Subject: Re: Sequence related to Goldbach Conjecture

Andrew Plewe wrote:
>     2  3  5  7  11 13 17 . . .
> 
> 2   4  5  7  9  13 15 19
> 3   5  6  8  10 14 16 20
> 5   7  8  10 12 16 18 22
> 7   9  10 12 14 18 20 24
> 11  13 14 16 18 22 24 28
> 13  15 16 18 20 24 26 30
> 17  19 20 22 24 28 30 34
> does the table above represent the most "efficient" table for 
> generating all the even integers greater than or equal to 4?



A perfectly efficient table can be constructed. Start with A000695:
    0,1,4,5,16,17,20,21,64,65,68,69,80,81,84,85,256,257,...

For each N in A000695, write 6N and 6N+1:
    0 1 6 7 24 25 30 31 96 97 102 103 120 121 126 127 384 385...

For each N in A000695, write 48N, 48N+2, 48N+4, 48N+12, 48N+14, and 48N+16:
    0 2 4 12 14 16
    48 50 52 60 62 64
    192 194 196 204 206 208
    ...

Let those two sequences be the index row and column of a sums table.
The interior of the table has each whole number exactly once.

 + |  0  1   6  7 24 25 30 31
---|-------------------------
 0 |  0  1   6  7 24 25 30 31
 2 |  2  3   8  9 26 27 32 33
 4 |  4  5  10 11 28 29 34 35
12 | 12 13  18 19 36 37 42 43
14 | 14 15  20 21 38 39 44 45
16 | 16 17  22 23 40 41 46 47
48 | 48 49 ...

Now, if you want just the even integers starting from 4, replace each index value X with 2X+2.

--
Don Reble  djr at nk.ca