[seqfan] Re: Squares with prime sums in rows and columns

[Christopher Gribble]

Hi Zak,

Here are some preliminary findings from generating each permutation of
1..N^2 and calculating the row and column sums of the corresponding square:

N	N^2 		  (N^2)!	No. of distinct row-column sum
combinations

1	  1 	             1		  1
2	  4 	            24		  3
3	  9 	        362880		820
4	 16	20922789888000		  ? (might take some time to
compute)

{1, 3, 820, ...} is not in the OEIS.

For N = 2, the distinct row-column sum combinations are

  3    4    6    7
  3    5    5    7
  4    5    5    6

Each combination is produced by 8 different squares.

For N = 3, the distinct row-column sum combinations are too numerous to list
here but I can supply them if anyone is interested.

The frequencies with which the different repetition counts occur for the
820 row-sum combinations are:

Repetition Count		Frequency
	  72			      82
	 144			     132
	 216			      99
	 288			      86
	 360			      50
	 432			      65
	 504			      32
	 576			      61
	 648			      35
	 720			      44
	 792			      11
	 864			      21
	 936			      13
	1008			      25
	1080			       8
	1152			       9
	1224			       3
	1296			       7
	1368			       2
	1440			       7
	1512			       5
	1584			       2
	1728			       3
	1800			       3
	1872			       2
	2016			       2
	2232			       1

Of the 820 combinations, there are 4 that consist of all primes:

						Repeat Count
   7   13   13   19   19   19			216
  11   11   11   17   17   23			288
  11   11   17   17   17   17			792
  13   13   13   13   19   19			360

Best regards,

Chris Gribble


-----Original Message-----
From: seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu]
On Behalf Of zak seidov
Sent: 22 October 2009 15:56
To: seqfaneu
Subject: [seqfan] Squares with prime sums in rows and columns

Dear seqfans,

is it new/interesting :-))
or old/dumb :-(


Thanks, Zak

Lexicographically least NxN squares of N^2 subsequent integers (not
necessarily starting with n=1) with prime sums in each row and each column
==============================================

1x1 square
rowsums={2}, colsums={2}
2
=

2x2 square
rowsums={3,7}, colsums={5,5}
1,2
4,3
==

3x3 square
rowsums={7,19,19}, colsums={13,13,19}
1,2,4
5,8,6
7,3,9
=====

4x4 square
rowsum={11,29,43,53}, colsum={23,31,41,41}
01,02,03,05
04,06,07,12
08,09,15,11
10,14,16,13
===========

5x5 square
rowsums={17,41,67,97,103}, colsums={53,53,59,71,89}
01,02,03,04,07
05,06,08,09,13
10,11,12,14,20
15,16,19,23,24
22,18,17,21,25
==============
      
6x6 square
rowsums={23,59,89,131,167,197}, colsums={97,103,109,113,113,131}    
01,02,03,04,05,08
06,07,09,10,11,16
12,13,14,15,17,18
19,20,21,22,23,26
24,25,28,29,30,31
35,36,34,33,27,32
=================
      
notes:  
1) in N=1 we start (and end!) with n=2
2) case N=2 has the only solution, while 
3) cases N>2 allow also other (any?) initial n's.
4) what about cases N>6?




      


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