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 |
2 | 43225617 | 25349939 | 2 |
3 | 30211581 | 25592569 | 12771406 | 3 |
4 | 43225617 | 19172862 | 0 | 6177077 | 4 |
5 | 18191459 | 15712903 | 9479138 | 9479135 | 15712921 | 5 |
6 | 19040332 | 12047669 | 10640385 | 11171249 | 13544900 | 2131021 | 6 |
7 | 12781299 | 13749712 | 13737861 | 4853228 | 13740548 | 4850538 | 4862370 | 7 |
8 | 30555423 | 16611259 | 0 | 4258185 | 12670194 | 2561603 | 0 | 1918892 | 8 |
9 | 30211581 | 8537616 | 4251772 | 0 | 8529457 | 4255744 | 0 | 8525496 | 4263890 | 9 |
10 | 11463324 | 7066239 | 7234479 | 2244663 | 8646678 | 6728135 | 8646664 | 2244659 | 7234472 | 7066243 | 10 |
11 | 7874504 | 8267234 | 3872975 | 8267231 | 8267236 | 8267247 | 3872953 | 3872970 | 3872983 | 8267228 | 3872995 |
A pictorial representation of the table above
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.