Sequence related to Goldbach Conjecture
Don Reble
djr at nk.ca
Wed Mar 30 20:48:17 CEST 2005
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
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