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 |
2 | 6095145 | 2108912 | 2 |
3 | 4372907 | 992073 | 2839077 | 3 |
4 | 6095145 | 739719 | 0 | 1369193 | 4 |
5 | 2636324 | 674719 | 2109144 | 2109139 | 674731 | 5 |
6 | 3062916 | 329695 | 2369851 | 1309991 | 662378 | 469226 | 6 |
7 | 1834678 | 1037252 | 1048503 | 1082503 | 1049483 | 1081524 | 1070114 | 7 |
8 | 5033124 | 172950 | 0 | 946136 | 1062021 | 566769 | 0 | 423057 | 8 |
9 | 4372907 | 325082 | 948476 | 0 | 334231 | 949929 | 0 | 332760 | 940672 | 9 |
10 | 1845109 | 161405 | 1611716 | 497448 | 513315 | 791215 | 513314 | 497428 | 1611691 | 161416 | 10 |
11 | 1108802 | 557841 | 861086 | 558056 | 558073 | 558044 | 861098 | 861082 | 861087 | 558066 | 860822 |
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.