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powerful numbers
A number n such that if prime p divides n then also p2 divides n. more

The first 600 powerful numbers :
1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, 1024, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1331, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1728, 1764, 1800, 1849, 1936, 1944, 2000, 2025, 2048, 2116, 2187, 2197, 2209, 2304, 2312, 2401, 2500, 2592, 2601, 2700, 2704, 2744, 2809, 2888, 2916, 3025, 3087, 3125, 3136, 3200, 3249, 3267, 3364, 3375, 3456, 3481, 3528, 3600, 3721, 3844, 3872, 3888, 3969, 4000, 4096, 4225, 4232, 4356, 4489, 4500, 4563, 4608, 4624, 4761, 4900, 4913, 5000, 5041, 5184, 5292, 5324, 5329, 5400, 5408, 5476, 5488, 5625, 5776, 5832, 5929, 6075, 6084, 6125, 6241, 6272, 6400, 6561, 6724, 6728, 6859, 6889, 6912, 7056, 7200, 7225, 7396, 7569, 7688, 7744, 7776, 7803, 7921, 8000, 8100, 8192, 8281, 8464, 8575, 8649, 8712, 8748, 8788, 8836, 9000, 9025, 9216, 9248, 9261, 9409, 9604, 9747, 9800, 9801, 10000, 10125, 10201, 10368, 10404, 10584, 10609, 10648, 10800, 10816, 10952, 10976, 11025, 11236, 11449, 11552, 11664, 11881, 11907, 11979, 12100, 12167, 12168, 12321, 12348, 12500, 12544, 12769, 12800, 12996, 13068, 13225, 13448, 13456, 13500, 13689, 13824, 13924, 14112, 14161, 14283, 14400, 14641, 14792, 14884, 15125, 15129, 15376, 15488, 15552, 15625, 15876, 16000, 16129, 16200, 16384, 16641, 16807, 16875, 16900, 16928, 17161, 17424, 17496, 17576, 17672, 17689, 17956, 18000, 18225, 18252, 18432, 18496, 18769, 19044, 19208, 19321, 19600, 19652, 19683, 19773, 19881, 20000, 20164, 20449, 20736, 20808, 21025, 21125, 21168, 21296, 21316, 21600, 21609, 21632, 21904, 21952, 22201, 22472, 22500, 22707, 22801, 23104, 23328, 23409, 23716, 24025, 24200, 24300, 24336, 24389, 24500, 24649, 24696, 24964, 25000, 25088, 25281, 25600, 25921, 25947, 25992, 26136, 26244, 26569, 26896, 26912, 27000, 27225, 27436, 27556, 27648, 27783, 27848, 27889, 28125, 28224, 28561, 28800, 28900, 29241, 29403, 29584, 29768, 29791, 29929, 30276, 30375, 30625, 30752, 30976, 31104, 31212, 31329, 31684, 31752, 32000, 32041, 32400, 32761, 32768, 33075, 33124, 33275, 33489, 33800, 33856, 34225, 34300, 34596, 34848, 34969, 34992, 35152, 35344, 35721, 35912, 35937, 36000, 36100, 36125, 36481, 36504, 36864, 36963, 36992, 37044, 37249, 37636, 38025, 38088, 38416, 38809, 38988, 39200, 39204, 39304, 39601, 40000, 40328, 40401, 40500, 40804, 41067, 41209, 41472, 41503, 41616, 42025, 42336, 42436, 42592, 42632, 42849, 42875, 43200, 43264, 43681, 43808, 43904, 44100, 44217, 44521, 44944, 45000, 45125, 45369, 45387, 45796, 46208, 46225, 46656, 47089, 47432, 47524, 47628, 47916, 47961, 48400, 48600, 48668, 48672, 48841, 49000, 49284, 49392, 49729, 49923, 49928, 50000, 50176, 50625, 50653, 51076, 51200, 51529, 51984, 52272, 52441, 52488, 52900, 53361, 53792, 53824, 54000, 54289, 54675, 54756, 54872, 54925, 55112, 55125, 55225, 55296, 55696, 56169, 56448, 56644, 57121, 57132, 57600, 57800, 57967, 58081, 58564, 59049, 59168, 59319, 59536, 59643, 60025, 60500, 60516, 60552, 61009, 61504, 61731, 61952, 62001, 62208, 62424, 62500, 63001, 63368, 63504, 64000, 64009, 64516, 64800, 64827, 65025, 65219, 65536, 66049, 66125, 66248, 66564, 67081, 67228, 67500, 67600, 67712, 68121, 68600, 68644, 68921, 69169, 69192, 69696, 69984, 70225, 70227, 70304, 70688, 70756, 71289, 71824, 72000, 72200, 72361, 72900, 73008, 73441, 73728, 73984, 74088, 74529, 75076, 75272, 75625, 75843, 76176, 76729, 76832, 77175, 77284, 77841, 77976, 78125, 78400, 78408, 78608, 78732, 78961, 79092, 79507, 79524, 80000, 80089, 80656, 81000, 81225, 81608, 81675, 81796, 82369, 82944, 83232, 83349, 83521, 84100, 84375, 84500, 84672, 84681, 84872, 85184, 85264, 85849, 86400, 86436, 86528, 87025, 87616, 87723, 87808, 88200, 88209, 88804, 89401, 89888, 90000, 90601, 90828, 91125, 91204, 91592, 91809, 92416, 93025, 93312, 93636, 93987, 94249, 94864, 95048.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 68575556 values, from 1 to 999999961946176).

n\r 0  1 
24322561725349939 2 
3302115812559256912771406 3 
4432256171917286206177077 4 
518191459157129039479138947913515712921 5 
619040332120476691064038511171249135449002131021 6 
712781299137497121373786148532281374054848505384862370 7 
830555423166112590425818512670194256160301918892 8 
93021158185376164251772085294574255744085254964263890 9 
1011463324706623972344792244663864667867281358646664224465972344727066243 10 
1178745048267234387297582672318267236826724738729533872970387298382672283872995

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.