A number of the form nk, for k > 1. more
The first 600 perfect powers :
1,
4,
8,
9,
16,
25,
27,
32,
36,
49,
64,
81,
100,
121,
125,
128,
144,
169,
196,
216,
225,
243,
256,
289,
324,
343,
361,
400,
441,
484,
512,
529,
576,
625,
676,
729,
784,
841,
900,
961,
1000,
1024,
1089,
1156,
1225,
1296,
1331,
1369,
1444,
1521,
1600,
1681,
1728,
1764,
1849,
1936,
2025,
2048,
2116,
2187,
2197,
2209,
2304,
2401,
2500,
2601,
2704,
2744,
2809,
2916,
3025,
3125,
3136,
3249,
3364,
3375,
3481,
3600,
3721,
3844,
3969,
4096,
4225,
4356,
4489,
4624,
4761,
4900,
4913,
5041,
5184,
5329,
5476,
5625,
5776,
5832,
5929,
6084,
6241,
6400,
6561,
6724,
6859,
6889,
7056,
7225,
7396,
7569,
7744,
7776,
7921,
8000,
8100,
8192,
8281,
8464,
8649,
8836,
9025,
9216,
9261,
9409,
9604,
9801,
10000,
10201,
10404,
10609,
10648,
10816,
11025,
11236,
11449,
11664,
11881,
12100,
12167,
12321,
12544,
12769,
12996,
13225,
13456,
13689,
13824,
13924,
14161,
14400,
14641,
14884,
15129,
15376,
15625,
15876,
16129,
16384,
16641,
16807,
16900,
17161,
17424,
17576,
17689,
17956,
18225,
18496,
18769,
19044,
19321,
19600,
19683,
19881,
20164,
20449,
20736,
21025,
21316,
21609,
21904,
21952,
22201,
22500,
22801,
23104,
23409,
23716,
24025,
24336,
24389,
24649,
24964,
25281,
25600,
25921,
26244,
26569,
26896,
27000,
27225,
27556,
27889,
28224,
28561,
28900,
29241,
29584,
29791,
29929,
30276,
30625,
30976,
31329,
31684,
32041,
32400,
32761,
32768,
33124,
33489,
33856,
34225,
34596,
34969,
35344,
35721,
35937,
36100,
36481,
36864,
37249,
37636,
38025,
38416,
38809,
39204,
39304,
39601,
40000,
40401,
40804,
41209,
41616,
42025,
42436,
42849,
42875,
43264,
43681,
44100,
44521,
44944,
45369,
45796,
46225,
46656,
47089,
47524,
47961,
48400,
48841,
49284,
49729,
50176,
50625,
50653,
51076,
51529,
51984,
52441,
52900,
53361,
53824,
54289,
54756,
54872,
55225,
55696,
56169,
56644,
57121,
57600,
58081,
58564,
59049,
59319,
59536,
60025,
60516,
61009,
61504,
62001,
62500,
63001,
63504,
64000,
64009,
64516,
65025,
65536,
66049,
66564,
67081,
67600,
68121,
68644,
68921,
69169,
69696,
70225,
70756,
71289,
71824,
72361,
72900,
73441,
73984,
74088,
74529,
75076,
75625,
76176,
76729,
77284,
77841,
78125,
78400,
78961,
79507,
79524,
80089,
80656,
81225,
81796,
82369,
82944,
83521,
84100,
84681,
85184,
85264,
85849,
86436,
87025,
87616,
88209,
88804,
89401,
90000,
90601,
91125,
91204,
91809,
92416,
93025,
93636,
94249,
94864,
95481,
96100,
96721,
97336,
97344,
97969,
98596,
99225,
99856,
100000,
100489,
101124,
101761,
102400,
103041,
103684,
103823,
104329,
104976,
105625,
106276,
106929,
107584,
108241,
108900,
109561,
110224,
110592,
110889,
111556,
112225,
112896,
113569,
114244,
114921,
115600,
116281,
116964,
117649,
118336,
119025,
119716,
120409,
121104,
121801,
122500,
123201,
123904,
124609,
125000,
125316,
126025,
126736,
127449,
128164,
128881,
129600,
130321,
131044,
131072,
131769,
132496,
132651,
133225,
133956,
134689,
135424,
136161,
136900,
137641,
138384,
139129,
139876,
140608,
140625,
141376,
142129,
142884,
143641,
144400,
145161,
145924,
146689,
147456,
148225,
148877,
148996,
149769,
150544,
151321,
152100,
152881,
153664,
154449,
155236,
156025,
156816,
157464,
157609,
158404,
159201,
160000,
160801,
161051,
161604,
162409,
163216,
164025,
164836,
165649,
166375,
166464,
167281,
168100,
168921,
169744,
170569,
171396,
172225,
173056,
173889,
174724,
175561,
175616,
176400,
177147,
177241,
178084,
178929,
179776,
180625,
181476,
182329,
183184,
184041,
184900,
185193,
185761,
186624,
187489,
188356,
189225,
190096,
190969,
191844,
192721,
193600,
194481,
195112,
195364,
196249,
197136,
198025,
198916,
199809,
200704,
201601,
202500,
203401,
204304,
205209,
205379,
206116,
207025,
207936,
208849,
209764,
210681,
211600,
212521,
213444,
214369,
215296,
216000,
216225,
217156,
218089,
219024,
219961,
220900,
221841,
222784,
223729,
224676,
225625,
226576,
226981,
227529,
228484,
229441,
230400,
231361,
232324,
233289,
234256,
235225,
236196,
237169,
238144,
238328,
239121,
240100,
241081,
242064,
243049,
244036,
245025,
246016,
247009,
248004,
248832,
249001,
250000,
250047,
251001,
252004,
253009,
254016,
255025,
256036,
257049,
258064,
259081,
260100,
261121,
262144,
263169,
264196,
265225,
266256,
267289,
268324,
269361,
270400,
271441,
272484,
273529,
274576,
274625,
275625,
276676,
277729,
278784.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 7108347 values, from 1 to 49999988518489).
n\r | 0 | 1 |
2 | 3554174 | 3554173 | 2 |
3 | 2369449 | 4726392 | 12506 | 3 |
4 | 3554174 | 3544797 | 0 | 9376 | 4 |
5 | 1421671 | 2835837 | 7503 | 7502 | 2835834 | 5 |
6 | 1184725 | 2363197 | 6254 | 1184724 | 2363195 | 6252 | 6 |
7 | 1015475 | 2036010 | 2020393 | 97 | 2020393 | 99 | 15880 | 7 |
8 | 1786407 | 3540108 | 0 | 4690 | 1767767 | 4689 | 0 | 4686 | 8 |
9 | 2369449 | 1583566 | 80 | 0 | 1571412 | 78 | 0 | 1571414 | 12348 | 9 |
10 | 710834 | 1417917 | 3751 | 3750 | 1417917 | 710837 | 1417920 | 3752 | 3752 | 1417917 | 10 |
11 | 646213 | 1289200 | 3361 | 1288970 | 1288972 | 1288972 | 3361 | 3362 | 3361 | 1288971 | 3604 |
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.