Squares for Scott

 
Squares for Scott
Hi Scott,
Our latest idea, inspired by your spirals (except we don’t spiral at all here !-)
 
We want to lexico-fill a quarter of an infinite board with distinct integers such that the [a(n)]^2 terms belonging to any square with a(n) in the top/left corner always sum up to a prime.
We fill a(n)’s square line by line, top to bottom and left to right.
 
The first square is thus:
 2  3
 4  8 with prime sum 17
The next square has 3 in its upper/left corner:
 2 .3  5  6
 4 .8  7  9
   10 11 12 with prime sum 71
The next square has 4 in its upper/left corner:
 2  3  5  6
.4  8  7  9
13 10 11 12
14 15 16 17
18 19 20 30 with prime sum 223
The next square has 5 in its upper/left corner:
 2  3 .5  6 21 22 23
 4  8 .7  9 24 25 26
13 10 11 12 27 28 29
14 15 16 17 31 32 33
18 19 20 30 34 35 40 with prime sum 563 (if I’m not wrong)
Etc.
 
The sequence that might be submitted to the OEIS would start with the integers read by the successive antidiagonals (starting in the NW corner of the ¼ board):
 
S = 2, 3, 4, 5, 8, 13, 6, 7, 10, 14, 21, 9, 11, 15, 18, 22, 24, 12, 16, 19,…
 
The above terms do not match anything in Neil’s table.
 
Best,
É.
 
 
 
 
 
 


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