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A figurate number of the form n2. more

The first 600 square numbers :
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3600, 3721, 3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241, 6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216, 9409, 9604, 9801, 10000, 10201, 10404, 10609, 10816, 11025, 11236, 11449, 11664, 11881, 12100, 12321, 12544, 12769, 12996, 13225, 13456, 13689, 13924, 14161, 14400, 14641, 14884, 15129, 15376, 15625, 15876, 16129, 16384, 16641, 16900, 17161, 17424, 17689, 17956, 18225, 18496, 18769, 19044, 19321, 19600, 19881, 20164, 20449, 20736, 21025, 21316, 21609, 21904, 22201, 22500, 22801, 23104, 23409, 23716, 24025, 24336, 24649, 24964, 25281, 25600, 25921, 26244, 26569, 26896, 27225, 27556, 27889, 28224, 28561, 28900, 29241, 29584, 29929, 30276, 30625, 30976, 31329, 31684, 32041, 32400, 32761, 33124, 33489, 33856, 34225, 34596, 34969, 35344, 35721, 36100, 36481, 36864, 37249, 37636, 38025, 38416, 38809, 39204, 39601, 40000, 40401, 40804, 41209, 41616, 42025, 42436, 42849, 43264, 43681, 44100, 44521, 44944, 45369, 45796, 46225, 46656, 47089, 47524, 47961, 48400, 48841, 49284, 49729, 50176, 50625, 51076, 51529, 51984, 52441, 52900, 53361, 53824, 54289, 54756, 55225, 55696, 56169, 56644, 57121, 57600, 58081, 58564, 59049, 59536, 60025, 60516, 61009, 61504, 62001, 62500, 63001, 63504, 64009, 64516, 65025, 65536, 66049, 66564, 67081, 67600, 68121, 68644, 69169, 69696, 70225, 70756, 71289, 71824, 72361, 72900, 73441, 73984, 74529, 75076, 75625, 76176, 76729, 77284, 77841, 78400, 78961, 79524, 80089, 80656, 81225, 81796, 82369, 82944, 83521, 84100, 84681, 85264, 85849, 86436, 87025, 87616, 88209, 88804, 89401, 90000, 90601, 91204, 91809, 92416, 93025, 93636, 94249, 94864, 95481, 96100, 96721, 97344, 97969, 98596, 99225, 99856, 100489, 101124, 101761, 102400, 103041, 103684, 104329, 104976, 105625, 106276, 106929, 107584, 108241, 108900, 109561, 110224, 110889, 111556, 112225, 112896, 113569, 114244, 114921, 115600, 116281, 116964, 117649, 118336, 119025, 119716, 120409, 121104, 121801, 122500, 123201, 123904, 124609, 125316, 126025, 126736, 127449, 128164, 128881, 129600, 130321, 131044, 131769, 132496, 133225, 133956, 134689, 135424, 136161, 136900, 137641, 138384, 139129, 139876, 140625, 141376, 142129, 142884, 143641, 144400, 145161, 145924, 146689, 147456, 148225, 148996, 149769, 150544, 151321, 152100, 152881, 153664, 154449, 155236, 156025, 156816, 157609, 158404, 159201, 160000, 160801, 161604, 162409, 163216, 164025, 164836, 165649, 166464, 167281, 168100, 168921, 169744, 170569, 171396, 172225, 173056, 173889, 174724, 175561, 176400, 177241, 178084, 178929, 179776, 180625, 181476, 182329, 183184, 184041, 184900, 185761, 186624, 187489, 188356, 189225, 190096, 190969, 191844, 192721, 193600, 194481, 195364, 196249, 197136, 198025, 198916, 199809, 200704, 201601, 202500, 203401, 204304, 205209, 206116, 207025, 207936, 208849, 209764, 210681, 211600, 212521, 213444, 214369, 215296, 216225, 217156, 218089, 219024, 219961, 220900, 221841, 222784, 223729, 224676, 225625, 226576, 227529, 228484, 229441, 230400, 231361, 232324, 233289, 234256, 235225, 236196, 237169, 238144, 239121, 240100, 241081, 242064, 243049, 244036, 245025, 246016, 247009, 248004, 249001, 250000, 251001, 252004, 253009, 254016, 255025, 256036, 257049, 258064, 259081, 260100, 261121, 262144, 263169, 264196, 265225, 266256, 267289, 268324, 269361, 270400, 271441, 272484, 273529, 274576, 275625, 276676, 277729, 278784, 279841, 280900, 281961, 283024, 284089, 285156, 286225, 287296, 288369, 289444, 290521, 291600, 292681, 293764, 294849, 295936, 297025, 298116, 299209, 300304, 301401, 302500, 303601, 304704, 305809, 306916, 308025, 309136, 310249, 311364, 312481, 313600, 314721, 315844, 316969, 318096, 319225, 320356, 321489, 322624, 323761, 324900, 326041, 327184, 328329, 329476, 330625, 331776, 332929, 334084, 335241, 336400, 337561, 338724, 339889, 341056, 342225, 343396, 344569, 345744, 346921, 348100, 349281, 350464, 351649, 352836, 354025, 355216, 356409, 357604, 358801, 360000.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 31622776 values, from 1 to 999999961946176).

n\r 0  1 
21581138815811388 2 
310540925210818510 3 
4158113881581138800 4 
56324555126491110012649110 5 
652704621054092505270463105409260 6 
74517539903507990350790903507900 7 
8790569415811388007905694000 8 
91054092570272830070272840070272840 9 
103162277632455500632455531622786324556006324555 10 
1128747975749595057495965749596574959600057495960

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.