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

Christopher Gribble chris.eveswell at virgin.net
Sun Oct 25 23:40:45 CET 2009

```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.

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```