[SeqFan] Benjamin Franklin's Magic Squares

[Paul D. Hanna]

Seqfans, 
     Do the following sets of numbers belong in the OEIS? 
There are too many magic squares in the universe to include, but perhaps 
this finite set has some historical significance that makes them worthy? 
 
First, I copy Benjamin Franklin's 8x8 magic square: 
52 61  4 13 20 29 36 45 
14  3 62 51 46 35 30 19 
53 60  5 12 21 28 37 44 
11  6 59 54 43 38 27 22 
55 58  7 10 23 26 39 42 
9  8 57 56 41 40 25 24 
50 63  2 15 18 31 34 47 
16  1 64 49 48 33 32 17 
where the the column and row sums equal 260. 
Should this be in the OEIS? 
 
At the bottom of this email, I copy Benjamin Franklin's 16x16 magic
square.
  
Related question - is it trivial or is it false in general that 
magic squares obey the following "rule"? 
The matrix powers of magic squares form number squares 
having the same sum for each of the columns and rows.
(Something tells me that this is not in true in general ... 
and depends on how the magic square is constructed.)  
  
Example:  the matrix square of Ben Franklin's 8x8 magic square is: 
[7794, 7378, 9522, 9106, 8946, 8530, 8370, 7954;
9266, 9746, 7154, 7634, 7858, 8338, 8562, 9042;
7954, 7602, 9298, 8946, 8850, 8498, 8402, 8050;
8786, 9074, 7826, 8114, 8146, 8434, 8466, 8754;
8274, 8050, 8850, 8626, 8658, 8434, 8466, 8242;
8466, 8626, 8274, 8434, 8338, 8498, 8402, 8562;
7474, 6930, 9970, 9426, 9138, 8594, 8306, 7762;
9586, 10194, 6706, 7314, 7666, 8274, 8626, 9234]
where the the column and row sums equal 67600 = 260^2. 
  
For these magic squares, all matrix powers produce non-unique elements 
in a number square having the same sum for each of the columns and rows. 
 
I wonder how many, say, 3X3 magic squares have matrix squares 
consisting of unique elements and have the same column and row sums? 
 
Just curious ... not very mathematically deep or serious ... 
      Paul 
----------------------------------

Benjamin Franklin's 16x16 magic square:

200 217 232 249   8  25  40  57  72  89 104 121 136 153 168 185
  58  39  26   7 250 231 218 199 186 167 154 135 122 103  90  71
198 219 230 251   6  27  38  59  70  91 102 123 134 155 166 187
  60  37  28   5 252 229 220 197 188 165 156 133 124 101  92  69
201 216 233 248   9  24  41  56  73  88 105 120 137 152 169 184
  55  42  23  10 247 234 215 202 183 170 151 138 119 106  87  74
203 214 235 246  11  22  43  54  75  86 107 118 139 150 171 182
  53  44  21  12 245 236 213 204 181 172 149 140 117 108  85  76
205 212 237 244  13  20  45  52  77  84 109 116 141 148 173 180
  51  46  19  14 243 238 211 206 179 174 147 142 115 110  83  78
207 210 239 242  15  18  47  50  79  82 111 114 143 146 175 178
  49  48  17  16 241 240 209 208 177 176 145 144 113 112  81  80
196 221 228 253   4  29  36  61  68  93 100 125 132 157 164 189
  62  35  30   3 254 227 222 195 190 163 158 131 126  99  94  67
194 223 226 255   2  31  34  63  66  95  98 127 130 159 162 191
  64  33  32   1 256 225 224 193 192 161 160 129 128  97  96  65

Source:
http://www.mathpages.com/home/kmath155.htm

Ps. the 16x16 magic square is slightly editted (from BF's original) by
the website owner.
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