Sequence related to Goldbach Conjecture

[Andrew Plewe]

Yes, this will work if you allow both even and odd values as inputs.  However, let us use the following rules:

a.) The only even integer allowed is 2 (because it is prime).  All others must be odd.
b.) We'll assume that since this is a truth table, input order shouldn't matter and thus the values along the top and down the
left-hand side should be the same.

Following these rules, constructing a table with greater efficiency is more difficult.  It's another way of asking if a set exists
that will generate all the even integers without using all the primes (or using a mixture of primes and odd composites), or as Ed
said "I assume that there is an effort to try to find the nonessential primes, in terms of Goldbach."  I don't know if such an
effort exists myself, but a quick google search on the phrase "non essential primes" returns a couple of papers about logic
reduction.  I'll chew through those and see if there's anything interesting.

	-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