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Achilles numbers
A number which is powerful, but not a perfect power. more

The first 600 Achilles numbers :
72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6075, 6125, 6272, 6728, 6912, 7200, 7688, 7803, 8575, 8712, 8748, 8788, 9000, 9248, 9747, 9800, 10125, 10368, 10584, 10800, 10952, 10976, 11552, 11907, 11979, 12168, 12348, 12500, 12800, 13068, 13448, 13500, 14112, 14283, 14792, 15125, 15488, 15552, 16000, 16200, 16875, 16928, 17496, 17672, 18000, 18252, 18432, 19208, 19652, 19773, 20000, 20808, 21125, 21168, 21296, 21600, 21632, 22472, 22707, 23328, 24200, 24300, 24500, 24696, 25000, 25088, 25947, 25992, 26136, 26912, 27436, 27648, 27783, 27848, 28125, 28800, 29403, 29768, 30375, 30752, 31104, 31212, 31752, 32000, 33075, 33275, 33800, 34300, 34848, 34992, 35152, 35912, 36000, 36125, 36504, 36963, 36992, 37044, 38088, 38988, 39200, 40328, 40500, 41067, 41472, 41503, 42336, 42592, 42632, 43200, 43808, 43904, 44217, 45000, 45125, 45387, 46208, 47432, 47628, 47916, 48600, 48668, 48672, 49000, 49392, 49923, 49928, 50000, 51200, 52272, 52488, 53792, 54000, 54675, 54925, 55112, 55125, 55296, 56448, 57132, 57800, 57967, 59168, 59643, 60500, 60552, 61731, 61952, 62208, 62424, 63368, 64800, 64827, 65219, 66125, 66248, 67228, 67500, 67712, 68600, 69192, 69984, 70227, 70304, 70688, 72000, 72200, 73008, 73728, 75272, 75843, 76832, 77175, 77976, 78408, 78608, 78732, 79092, 80000, 81000, 81608, 81675, 83232, 83349, 84375, 84500, 84672, 84872, 86400, 86528, 87723, 87808, 88200, 89888, 90828, 91592, 93312, 93987, 95048, 95256, 95832, 96800, 97200, 97556, 98000, 98568, 98784, 99127, 100352, 100467, 102152, 103788, 103968, 104544, 105125, 105800, 107163, 107648, 107653, 107811, 108000, 109503, 109512, 109744, 111132, 111392, 112500, 113288, 114075, 114264, 115200, 117128, 117612, 119072, 119164, 120125, 121000, 121032, 121203, 121500, 122825, 123008, 123823, 124416, 124848, 127008, 128000, 128547, 129032, 132300, 133100, 133128, 134456, 135000, 135200, 136107, 136125, 137200, 137288, 139392, 139968, 141512, 143648, 143748, 143883, 144000, 144500, 145800, 146016, 147852, 147968, 148176, 150152, 151263, 151875, 152352, 153125, 154568, 155952, 156800, 157216, 158184, 159048, 160083, 161312, 162000, 163592, 164268, 165375, 165888, 166012, 168200, 168507, 169000, 169344, 170368, 170528, 171125, 171475, 171500, 172800, 172872, 175232, 176868, 177608, 177957, 180000, 180500, 181447, 181548, 181656, 182408, 184832, 186003, 187272, 189728, 190125, 190512, 191664, 192200, 194400, 194672, 194688, 195075, 196000, 197192, 197568, 199692, 199712, 200000, 202248, 202612, 204363, 204800, 207368, 207576, 209088, 209952, 210125, 212552, 213867, 214375, 215168, 217800, 218700, 219488, 219501, 219700, 220448, 220500, 221184, 222264, 223112, 223587, 224939, 225000, 225792, 228488, 228528, 231125, 231200, 231525, 231868, 233523, 233928, 235224, 236672, 237276, 238572, 239432, 240737, 242000, 242208, 243000, 243675, 245000, 246924, 247808, 249696, 250632, 253125, 253472, 254043, 256000, 256328, 259200, 259308, 260876, 262088, 264500, 264600, 264627, 264992, 265837, 266200, 267912, 268119, 268912, 270000, 270848, 273375, 273800, 274400, 275427, 275684, 276125, 276768, 279752, 280908, 281216, 282752, 285768, 286443, 288000, 288463, 288800, 289000, 291848, 292032, 294912, 295704, 296352, 297675, 297992, 299475, 300125, 301088, 303372, 304175, 304200, 307328, 308700, 309123, 310472, 311904, 312500, 313632, 314928, 316368, 316808, 318028, 320000, 320787, 323208, 323433, 324000, 325125, 326432, 326700, 328536, 329623, 329672, 332024, 332667, 332928, 333396, 336091, 336200, 337500, 338000, 338688, 339488, 340736, 342792, 344763, 345600, 346112, 349448, 350892, 351125, 351232, 352800, 353736, 356168, 357075, 359552, 361000, 362952, 363096, 363312, 364500, 366368, 369603, 369800, 373527, 375948, 376712, 378125, 380192, 381024, 382347, 383328, 383688, 384659, 385875, 387200, 388800, 389344, 390224, 390728, 392000, 394272, 395136, 395307, 396508, 397832, 397953, 399384, 400000, 401408, 401868, 405000, 406125, 408375, 408483, 408608, 412232, 415152, 415292, 415872, 418176, 419528, 420175, 420500, 423200, 426888, 428652, 430592, 430612, 431244, 432000, 434312, 435125, 435483, 437400, 438012, 438048, 439400, 441000, 441800, 442368, 444528, 445568, 449307, 449352, 450000, 453152, 453789, 455877, 456300, 456968, 457056, 460800, 463347, 463736, 464648.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 8204057 values, from 72 to 50000000000000).

n\r 0  1 
260951452108912 2 
343729079920732839077 3 
4609514573971901369193 4 
5263632467471921091442109139674731 5 
6306291632969523698511309991662378469226 6 
71834678103725210485031082503104948310815241070114 7 
85033124172950094613610620215667690423057 8 
9437290732508294847603342319499290332760940672 9 
10184510916140516117164974485133157912155133144974281611691161416 10 
111108802557841861086558056558073558044861098861082861087558066860822

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.