A number of the form bk ± 1 with b, k > 1 more
The first 600 Cunningham numbers :
3,
5,
7,
8,
9,
10,
15,
17,
24,
26,
28,
31,
33,
35,
37,
48,
50,
63,
65,
80,
82,
99,
101,
120,
122,
124,
126,
127,
129,
143,
145,
168,
170,
195,
197,
215,
217,
224,
226,
242,
244,
255,
257,
288,
290,
323,
325,
342,
344,
360,
362,
399,
401,
440,
442,
483,
485,
511,
513,
528,
530,
575,
577,
624,
626,
675,
677,
728,
730,
783,
785,
840,
842,
899,
901,
960,
962,
999,
1001,
1023,
1025,
1088,
1090,
1155,
1157,
1224,
1226,
1295,
1297,
1330,
1332,
1368,
1370,
1443,
1445,
1520,
1522,
1599,
1601,
1680,
1682,
1727,
1729,
1763,
1765,
1848,
1850,
1935,
1937,
2024,
2026,
2047,
2049,
2115,
2117,
2186,
2188,
2196,
2198,
2208,
2210,
2303,
2305,
2400,
2402,
2499,
2501,
2600,
2602,
2703,
2705,
2743,
2745,
2808,
2810,
2915,
2917,
3024,
3026,
3124,
3126,
3135,
3137,
3248,
3250,
3363,
3365,
3374,
3376,
3480,
3482,
3599,
3601,
3720,
3722,
3843,
3845,
3968,
3970,
4095,
4097,
4224,
4226,
4355,
4357,
4488,
4490,
4623,
4625,
4760,
4762,
4899,
4901,
4912,
4914,
5040,
5042,
5183,
5185,
5328,
5330,
5475,
5477,
5624,
5626,
5775,
5777,
5831,
5833,
5928,
5930,
6083,
6085,
6240,
6242,
6399,
6401,
6560,
6562,
6723,
6725,
6858,
6860,
6888,
6890,
7055,
7057,
7224,
7226,
7395,
7397,
7568,
7570,
7743,
7745,
7775,
7777,
7920,
7922,
7999,
8001,
8099,
8101,
8191,
8193,
8280,
8282,
8463,
8465,
8648,
8650,
8835,
8837,
9024,
9026,
9215,
9217,
9260,
9262,
9408,
9410,
9603,
9605,
9800,
9802,
9999,
10001,
10200,
10202,
10403,
10405,
10608,
10610,
10647,
10649,
10815,
10817,
11024,
11026,
11235,
11237,
11448,
11450,
11663,
11665,
11880,
11882,
12099,
12101,
12166,
12168,
12320,
12322,
12543,
12545,
12768,
12770,
12995,
12997,
13224,
13226,
13455,
13457,
13688,
13690,
13823,
13825,
13923,
13925,
14160,
14162,
14399,
14401,
14640,
14642,
14883,
14885,
15128,
15130,
15375,
15377,
15624,
15626,
15875,
15877,
16128,
16130,
16383,
16385,
16640,
16642,
16806,
16808,
16899,
16901,
17160,
17162,
17423,
17425,
17575,
17577,
17688,
17690,
17955,
17957,
18224,
18226,
18495,
18497,
18768,
18770,
19043,
19045,
19320,
19322,
19599,
19601,
19682,
19684,
19880,
19882,
20163,
20165,
20448,
20450,
20735,
20737,
21024,
21026,
21315,
21317,
21608,
21610,
21903,
21905,
21951,
21953,
22200,
22202,
22499,
22501,
22800,
22802,
23103,
23105,
23408,
23410,
23715,
23717,
24024,
24026,
24335,
24337,
24388,
24390,
24648,
24650,
24963,
24965,
25280,
25282,
25599,
25601,
25920,
25922,
26243,
26245,
26568,
26570,
26895,
26897,
26999,
27001,
27224,
27226,
27555,
27557,
27888,
27890,
28223,
28225,
28560,
28562,
28899,
28901,
29240,
29242,
29583,
29585,
29790,
29792,
29928,
29930,
30275,
30277,
30624,
30626,
30975,
30977,
31328,
31330,
31683,
31685,
32040,
32042,
32399,
32401,
32760,
32762,
32767,
32769,
33123,
33125,
33488,
33490,
33855,
33857,
34224,
34226,
34595,
34597,
34968,
34970,
35343,
35345,
35720,
35722,
35936,
35938,
36099,
36101,
36480,
36482,
36863,
36865,
37248,
37250,
37635,
37637,
38024,
38026,
38415,
38417,
38808,
38810,
39203,
39205,
39303,
39305,
39600,
39602,
39999,
40001,
40400,
40402,
40803,
40805,
41208,
41210,
41615,
41617,
42024,
42026,
42435,
42437,
42848,
42850,
42874,
42876,
43263,
43265,
43680,
43682,
44099,
44101,
44520,
44522,
44943,
44945,
45368,
45370,
45795,
45797,
46224,
46226,
46655,
46657,
47088,
47090,
47523,
47525,
47960,
47962,
48399,
48401,
48840,
48842,
49283,
49285,
49728,
49730,
50175,
50177,
50624,
50626,
50652,
50654,
51075,
51077,
51528,
51530,
51983,
51985,
52440,
52442,
52899,
52901,
53360,
53362,
53823,
53825,
54288,
54290,
54755,
54757,
54871,
54873,
55224,
55226,
55695,
55697,
56168,
56170,
56643,
56645,
57120,
57122,
57599,
57601,
58080,
58082,
58563,
58565,
59048,
59050,
59318,
59320,
59535,
59537,
60024,
60026,
60515,
60517,
61008,
61010,
61503,
61505,
62000,
62002,
62499,
62501,
63000,
63002,
63503,
63505,
63999,
64001,
64008,
64010,
64515,
64517,
65024,
65026,
65535,
65537,
66048,
66050,
66563,
66565,
67080.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 63447179 values, from 3 to 999999961946177).
n\r | 0 | 1 |
2 | 31723587 | 31723592 | 2 |
3 | 21149058 | 10608260 | 31689861 | 3 |
4 | 15861794 | 15861796 | 15861793 | 15861796 | 4 |
5 | 25338397 | 6364954 | 12689437 | 12689438 | 6364953 | 5 |
6 | 10574527 | 5304132 | 15844932 | 10574531 | 5304128 | 15844929 | 6 |
7 | 9120834 | 13567176 | 9077986 | 18070473 | 343 | 9078255 | 4532112 | 7 |
8 | 15836497 | 7956102 | 15836500 | 7905694 | 25297 | 7905694 | 25293 | 7956102 | 8 |
9 | 7093978 | 10574669 | 7060523 | 7027540 | 136 | 7027403 | 7027540 | 33455 | 17601935 | 9 |
10 | 12669196 | 3182475 | 6344716 | 6344719 | 3182477 | 12669201 | 3182479 | 6344721 | 6344719 | 3182476 | 10 |
11 | 5768620 | 2893069 | 11517709 | 5767751 | 11517285 | 5767751 | 5767753 | 18218 | 5767754 | 18664 | 8642605 |
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.