A composite number n for which there exists a k such that φ(n) divides (n-1)k. more
The first 600 k-Lehmer numbers :
15,
51,
85,
91,
133,
247,
255,
259,
435,
451,
481,
511,
561,
595,
679,
703,
763,
771,
949,
1105,
1111,
1141,
1261,
1285,
1351,
1387,
1417,
1615,
1695,
1729,
1843,
1891,
2047,
2071,
2091,
2119,
2431,
2465,
2509,
2701,
2761,
2821,
2955,
3031,
3097,
3145,
3277,
3367,
3409,
3589,
3655,
3667,
3855,
4033,
4039,
4141,
4369,
4411,
4681,
4795,
4921,
5083,
5151,
5383,
5461,
5551,
5611,
5629,
5713,
6031,
6205,
6331,
6601,
6643,
6735,
7051,
7081,
7141,
7471,
7501,
7735,
7957,
8071,
8119,
8227,
8245,
8401,
8481,
8695,
8827,
8911,
8995,
9061,
9079,
9139,
9211,
9253,
9265,
9367,
9605,
9709,
9919,
9997,
10213,
10291,
10573,
10585,
10795,
10963,
11041,
11155,
11305,
11899,
12403,
12801,
12901,
13021,
13107,
13333,
13651,
13741,
13747,
13855,
13981,
14089,
14491,
14497,
14611,
14701,
14833,
14911,
14989,
15051,
15181,
15211,
15811,
15841,
16021,
16297,
16405,
16441,
16471,
16531,
16705,
16745,
16771,
16861,
17563,
17611,
17733,
17767,
18019,
18031,
18151,
18631,
18721,
18745,
18907,
18967,
19345,
19669,
19951,
20419,
20451,
20595,
20995,
21037,
21349,
21679,
21845,
21907,
21931,
22015,
22261,
22321,
22359,
23001,
23281,
23959,
24199,
24415,
24643,
24661,
24727,
24871,
25123,
25141,
25351,
25669,
26281,
26335,
26467,
26599,
26923,
27223,
27331,
27511,
27721,
28231,
28453,
28645,
28939,
29341,
30481,
30583,
30889,
31171,
31417,
31459,
31609,
31611,
31621,
32215,
32407,
32551,
32691,
33001,
33709,
34441,
34861,
35113,
35371,
35551,
35881,
36091,
36391,
36499,
36661,
36751,
36805,
37231,
37921,
37969,
38165,
38503,
39091,
39403,
39491,
39817,
39831,
39865,
40261,
40501,
40885,
40921,
41041,
41245,
41395,
41881,
42001,
42121,
42127,
42217,
42661,
42799,
43165,
43387,
43435,
43621,
44011,
44317,
44941,
45551,
45811,
45991,
46657,
47055,
47197,
47239,
47545,
47671,
47989,
48133,
48709,
49045,
49051,
49141,
49155,
49231,
49267,
49471,
49567,
49771,
49987,
50251,
50557,
50737,
50881,
51319,
52429,
52507,
52633,
52801,
53083,
53131,
53983,
54223,
54691,
54741,
54811,
55423,
55471,
55537,
55651,
55699,
55831,
55969,
56137,
56551,
56661,
56797,
57205,
57589,
57715,
59641,
59755,
61249,
62745,
62893,
63139,
63631,
63973,
64201,
64345,
64741,
64771,
64855,
65311,
65365,
65535,
65641,
65683,
66991,
67771,
68101,
68191,
68251,
68341,
68619,
69231,
69601,
69751,
69781,
69979,
70579,
71905,
72535,
72583,
72631,
73261,
73555,
73801,
73891,
74011,
74023,
74593,
74665,
75055,
75361,
76627,
76741,
76921,
77161,
77871,
78013,
78403,
79003,
79381,
80665,
80971,
81631,
81651,
81691,
82411,
83011,
83293,
83569,
83665,
83731,
83821,
84169,
85471,
85591,
85695,
85765,
86023,
86731,
87061,
87123,
88357,
88453,
88561,
88621,
88831,
90751,
90973,
91001,
92053,
92491,
92701,
92833,
92929,
93031,
93805,
93961,
93991,
94051,
94657,
94681,
94831,
95941,
96139,
96691,
96985,
97351,
97567,
97921,
98005,
98623,
98671,
98881,
99499,
100651,
101101,
101311,
101935,
102277,
103441,
104653,
104923,
105001,
106141,
106507,
106951,
107185,
107929,
108691,
108781,
109291,
109861,
110245,
110371,
110677,
111055,
111361,
111841,
112141,
112147,
113201,
113401,
113611,
113821,
114589,
114835,
115231,
115627,
115921,
116705,
116935,
117181,
117895,
118027,
118405,
119461,
120171,
120289,
120445,
121435,
122221,
122461,
122479,
123151,
124015,
124369,
124645,
125341,
125347,
125401,
125563,
125581,
125677,
125809,
126133,
126217,
126331,
126673,
126701,
127723,
127909,
128143,
128371,
129031,
129795,
130351,
130381,
131047,
131461,
131821,
132595,
132769,
133141,
134521,
134539,
134797,
135679,
135931,
136741,
136981,
137149,
137311,
137449,
137971,
138481,
139231,
140065,
140811,
140911,
141373,
141523,
141571,
141661,
142519,
142681,
143893,
144673,
144841,
145027,
145351,
145993,
146611,
146791,
146965,
147109,
147259,
147511,
147967,
148291,
148417,
148831,
149443,
150691,
151183,
151747,
151887,
152551,
153349,
153595,
154687,
155149,
155365,
156655,
156877,
157641,
159031,
159259,
159757,
159895,
160147,
160291,
160891,
161653,
161701,
162401,
162871,
162955,
163885,
163891,
164041,
164761,
167713,
167761,
168103,
168421,
170431,
170905,
171205,
171445,
172081,
172291,
172501,
173419,
176011,
176035,
176149,
176437,
180115,
180851,
180991,
181351,
181579,
182527,
182533,
183061,
183157,
184255,
184891,
186961,
187867,
187939,
188191.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 2103055 values, from 15 to 999998343631).
n\r | 0 | 1 |
2 | 0 | 2103055 | 2 |
3 | 10278 | 2081983 | 10794 | 3 |
4 | 0 | 1071130 | 0 | 1031925 | 4 |
5 | 129045 | 1636385 | 115058 | 111672 | 110895 | 5 |
6 | 0 | 2081983 | 0 | 10278 | 0 | 10794 | 6 |
7 | 282560 | 1036864 | 155029 | 159771 | 155379 | 155542 | 157910 | 7 |
8 | 0 | 539531 | 0 | 515924 | 0 | 531599 | 0 | 516001 | 8 |
9 | 0 | 722074 | 3615 | 5095 | 679118 | 3621 | 5183 | 680791 | 3558 | 9 |
10 | 0 | 1636385 | 0 | 111672 | 0 | 129045 | 0 | 115058 | 0 | 110895 | 10 |
11 | 248283 | 563510 | 143684 | 142576 | 141943 | 143488 | 144075 | 143591 | 144453 | 143232 | 144220 |
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.