A number whose digits in a given base are all equal to 1. more
The first 600 repunits :
7,
13,
15,
21,
31,
40,
43,
57,
63,
73,
85,
91,
111,
121,
127,
133,
156,
157,
183,
211,
241,
255,
259,
273,
307,
341,
343,
364,
381,
400,
421,
463,
507,
511,
553,
585,
601,
651,
703,
757,
781,
813,
820,
871,
931,
993,
1023,
1057,
1093,
1111,
1123,
1191,
1261,
1333,
1365,
1407,
1464,
1483,
1555,
1561,
1641,
1723,
1807,
1885,
1893,
1981,
2047,
2071,
2163,
2257,
2353,
2380,
2451,
2551,
2653,
2757,
2801,
2863,
2955,
2971,
3081,
3193,
3280,
3307,
3423,
3541,
3616,
3661,
3783,
3906,
3907,
4033,
4095,
4161,
4291,
4369,
4423,
4557,
4681,
4693,
4831,
4971,
5113,
5220,
5257,
5403,
5461,
5551,
5701,
5853,
6007,
6163,
6175,
6321,
6481,
6643,
6807,
6973,
7141,
7240,
7311,
7381,
7483,
7657,
7833,
8011,
8191,
8373,
8421,
8557,
8743,
8931,
9121,
9313,
9331,
9507,
9703,
9724,
9841,
9901,
10101,
10303,
10507,
10713,
10921,
11111,
11131,
11155,
11343,
11557,
11773,
11991,
12211,
12433,
12657,
12720,
12883,
13111,
13341,
13573,
13807,
14043,
14281,
14425,
14521,
14763,
15007,
15253,
15501,
15751,
16003,
16105,
16257,
16276,
16383,
16513,
16771,
17031,
17293,
17557,
17823,
18091,
18279,
18361,
18633,
18907,
19183,
19461,
19531,
19608,
19741,
20023,
20307,
20440,
20593,
20881,
21171,
21463,
21757,
21845,
22053,
22351,
22621,
22651,
22765,
22953,
23257,
23563,
23871,
24181,
24493,
24807,
25123,
25260,
25441,
25761,
26083,
26407,
26733,
27061,
27391,
27723,
27931,
28057,
28393,
28731,
29071,
29413,
29524,
29757,
30103,
30451,
30784,
30801,
30941,
31153,
31507,
31863,
32221,
32581,
32767,
32943,
33307,
33673,
33825,
34041,
34411,
34783,
35157,
35533,
35911,
36291,
36673,
37057,
37060,
37443,
37449,
37831,
38221,
38613,
39007,
39403,
39801,
40201,
40495,
40603,
41007,
41371,
41413,
41821,
42231,
42643,
43057,
43473,
43891,
44136,
44311,
44733,
45157,
45583,
46011,
46441,
46873,
47307,
47743,
47989,
48181,
48621,
49063,
49507,
49953,
50401,
50851,
51303,
51757,
52060,
52213,
52671,
53131,
53593,
54057,
54241,
54523,
54991,
55461,
55933,
55987,
56355,
56407,
56883,
57361,
57841,
58323,
58807,
59293,
59781,
60271,
60763,
60880,
61257,
61753,
62251,
62751,
63253,
63757,
64263,
64771,
65281,
65535,
65641,
65793,
66307,
66430,
66823,
67341,
67861,
68383,
68907,
69433,
69905,
69961,
70491,
70644,
71023,
71557,
72093,
72631,
73171,
73713,
74257,
74803,
75351,
75895,
75901,
76453,
77007,
77563,
78121,
78681,
79243,
79807,
80373,
80941,
81400,
81511,
82083,
82657,
83233,
83811,
84391,
84973,
85557,
86143,
86731,
87165,
87321,
87381,
87913,
88507,
88573,
88741,
89103,
89701,
90301,
90903,
91507,
92113,
92721,
93196,
93331,
93943,
94557,
95173,
95791,
96411,
97033,
97656,
97657,
98283,
98911,
99499,
99541,
100173,
100807,
101443,
102081,
102721,
103363,
104007,
104653,
105301,
105951,
106080,
106603,
107257,
107913,
108571,
109231,
109893,
110557,
111111,
111151,
111223,
111891,
112561,
112945,
113233,
113907,
114583,
115261,
115941,
116623,
117307,
117993,
118681,
119371,
120063,
120100,
120757,
121453,
122151,
122851,
123553,
124257,
124963,
125671,
126381,
127093,
127551,
127807,
128523,
129241,
129961,
130683,
131071,
131407,
132133,
132861,
133591,
134323,
135057,
135304,
135793,
136531,
137257,
137271,
137561,
138013,
138757,
139503,
140251,
141001,
141753,
142507,
143263,
143365,
144021,
144781,
145543,
146307,
147073,
147841,
148611,
149383,
150157,
150933,
151711,
151740,
152491,
153273,
154057,
154843,
155631,
156421,
157213,
158007,
158803,
159601,
160401,
160435,
161203,
162007,
162813,
163621,
164431,
165243,
166057,
166873,
167691,
168421,
168511,
169333,
169456,
170157,
170983,
171811,
172641,
173473,
174307,
175143,
175981,
176821,
177156,
177663,
178507,
178809,
179353,
180201,
181051,
181903,
182757,
183613,
184471,
185331,
186193,
187057,
187923,
188500,
188791,
189661,
190533,
191407,
192283,
193161,
194041,
194923,
195807,
196693,
197581,
198471,
198535,
199363,
200257,
201153,
202051,
202951,
203853,
204205,
204757,
205663,
206571,
207481,
208393,
208920,
209307,
210223,
211141,
212061,
212983,
213907,
214833,
215761,
216691,
217623,
218557,
219493,
219661,
220431,
221371,
222313,
223257,
224203,
225151,
226101,
227053,
228007,
228963,
229921,
230764,
230881,
231843,
232807,
233773,
234741,
235711,
236683,
237657,
238633,
239611,
240591.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 31730131 values, from 7 to 999999993568953).
n\r | 0 | 1 |
2 | 50621 | 31679510 | 2 |
3 | 10575057 | 21153137 | 1937 | 3 |
4 | 50356 | 15841172 | 265 | 15838338 | 4 |
5 | 61467 | 12674293 | 6324702 | 12649437 | 20232 | 5 |
6 | 17042 | 21119582 | 24 | 10558015 | 33555 | 1913 | 6 |
7 | 9050214 | 9080086 | 15115 | 9035938 | 15122 | 15934 | 4517722 | 7 |
8 | 25214 | 7920596 | 133 | 7919152 | 25142 | 7920576 | 132 | 7919186 | 8 |
9 | 11507 | 7039970 | 645 | 10552074 | 7073206 | 646 | 11476 | 7039961 | 646 | 9 |
10 | 30164 | 12664068 | 2 | 12649318 | 10111 | 31303 | 10225 | 6324700 | 119 | 10121 | 10 |
11 | 11292 | 5760293 | 5767825 | 5749713 | 27832 | 674 | 18950 | 5759246 | 9270 | 2875345 | 5749691 |
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.