Sequence related to Goldbach Conjecture

[Andrew Plewe]

Let us construct a truth table using primes, where the values in the table consist of the sums of the primes, like this:

    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
.
.
.


Now, we count the number of first-occurances of the even numbers in each column, above the diagonal, starting with the first column
on the left.  We get this sequence:

1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 5, 4, 5, 3, 4, 3, 3, 7, 3, etc., or

(1) = 1, because 4 first occurs in column 1,
(2) = 1, because 6 first occurs in column 2,
(3) = 2, because 8 and 10 first occur in column 3,
and so on.


I will figure out more terms for this sequence and submit it this evening.  I do have a question that I thought of while making this
sequence -- does the table above represent the most "efficient" table for generating all the even integers greater than or equal to
4?  By that I mean is there a table that can have fewer columns and rows and yet still produce all the even integers.  I'm also
assuming a rule that all the row and column values must be either odd or prime (thus allowing for the inclusion of 2).  The answer
partially depends on the truth of the Goldbach Conjecture (if it's not true, the table above doesn't generate all even numbers
greater than 4).  However, even if it is true that doesn't necessarily mean that constructing the table using all the primes is the
most efficient method (unless, of course, one could prove otherwise...).


	-Andrew Plewe-