A number n > 3 which divides kn-2 - k for all 1 < k < n relatively prime to n. more
The first 600 D-numbers :
9,
15,
21,
33,
39,
51,
57,
63,
69,
87,
93,
111,
123,
129,
141,
159,
177,
183,
195,
201,
213,
219,
237,
249,
267,
291,
303,
309,
315,
321,
327,
339,
381,
393,
399,
411,
417,
447,
453,
471,
489,
501,
519,
537,
543,
573,
579,
591,
597,
633,
669,
681,
687,
693,
699,
717,
723,
753,
771,
789,
807,
813,
819,
831,
843,
849,
879,
921,
933,
939,
951,
993,
1011,
1023,
1041,
1047,
1059,
1077,
1101,
1119,
1137,
1149,
1167,
1191,
1203,
1227,
1257,
1263,
1293,
1299,
1317,
1329,
1347,
1371,
1383,
1389,
1401,
1437,
1443,
1461,
1473,
1497,
1509,
1527,
1563,
1569,
1623,
1641,
1671,
1683,
1689,
1707,
1713,
1731,
1761,
1779,
1797,
1803,
1821,
1839,
1851,
1857,
1893,
1923,
1929,
1935,
1941,
1953,
1959,
1977,
1983,
2019,
2031,
2049,
2073,
2103,
2127,
2157,
2181,
2199,
2217,
2229,
2253,
2271,
2283,
2307,
2319,
2361,
2391,
2427,
2433,
2463,
2469,
2481,
2487,
2517,
2559,
2571,
2577,
2589,
2631,
2643,
2649,
2661,
2721,
2733,
2757,
2787,
2811,
2823,
2841,
2859,
2901,
2913,
2931,
2949,
2973,
2991,
3003,
3027,
3039,
3057,
3063,
3093,
3099,
3117,
3147,
3153,
3183,
3189,
3207,
3261,
3273,
3279,
3291,
3309,
3315,
3327,
3351,
3369,
3387,
3453,
3459,
3489,
3513,
3543,
3561,
3579,
3603,
3639,
3651,
3669,
3687,
3693,
3711,
3747,
3777,
3831,
3837,
3843,
3849,
3867,
3873,
3891,
3903,
3909,
3921,
3957,
3963,
3981,
4083,
4095,
4101,
4119,
4143,
4197,
4227,
4269,
4281,
4287,
4299,
4317,
4341,
4353,
4359,
4377,
4413,
4443,
4449,
4461,
4467,
4479,
4497,
4533,
4569,
4593,
4623,
4629,
4647,
4659,
4677,
4701,
4713,
4737,
4749,
4791,
4803,
4821,
4827,
4839,
4857,
4863,
4881,
4911,
4971,
4989,
5001,
5007,
5079,
5091,
5097,
5127,
5163,
5169,
5187,
5199,
5223,
5241,
5259,
5277,
5331,
5349,
5361,
5367,
5403,
5433,
5469,
5493,
5541,
5583,
5601,
5613,
5619,
5631,
5637,
5667,
5703,
5721,
5739,
5793,
5799,
5847,
5853,
5919,
5937,
5961,
5979,
5991,
5997,
6009,
6033,
6051,
6081,
6087,
6117,
6159,
6189,
6207,
6243,
6249,
6261,
6267,
6297,
6333,
6339,
6387,
6393,
6411,
6423,
6429,
6459,
6483,
6537,
6609,
6621,
6639,
6663,
6711,
6717,
6729,
6753,
6801,
6807,
6819,
6843,
6861,
6879,
6891,
6927,
6933,
6999,
7017,
7023,
7041,
7053,
7071,
7113,
7131,
7143,
7149,
7167,
7179,
7197,
7233,
7251,
7269,
7311,
7323,
7341,
7377,
7395,
7401,
7419,
7431,
7509,
7563,
7593,
7617,
7629,
7647,
7653,
7671,
7737,
7773,
7779,
7827,
7851,
7863,
7899,
7941,
7971,
7977,
7989,
8013,
8031,
8049,
8061,
8067,
8079,
8097,
8121,
8133,
8139,
8157,
8187,
8193,
8223,
8247,
8259,
8301,
8331,
8367,
8373,
8391,
8403,
8409,
8457,
8463,
8499,
8511,
8529,
8553,
8571,
8583,
8637,
8661,
8691,
8709,
8727,
8751,
8781,
8817,
8859,
8871,
8889,
8907,
8913,
8997,
9003,
9033,
9057,
9069,
9111,
9123,
9147,
9183,
9201,
9237,
9249,
9267,
9327,
9357,
9363,
9411,
9489,
9501,
9507,
9543,
9561,
9573,
9603,
9609,
9627,
9651,
9663,
9687,
9753,
9759,
9771,
9777,
9813,
9897,
9903,
9921,
9939,
9957,
9969,
9987,
9993,
10029,
10041,
10077,
10083,
10113,
10119,
10167,
10173,
10221,
10239,
10299,
10347,
10371,
10383,
10389,
10401,
10407,
10473,
10497,
10533,
10551,
10581,
10587,
10599,
10617,
10623,
10641,
10671,
10677,
10713,
10743,
10749,
10779,
10815,
10821,
10839,
10851,
10869,
10893,
10911,
10929,
10977,
11013,
11019,
11031,
11073,
11091,
11103,
11127,
11157,
11181,
11199,
11217,
11283,
11301,
11307,
11337,
11379,
11391,
11409,
11463,
11469,
11499,
11541,
11553,
11559,
11589,
11631,
11643,
11667,
11721,
11733,
11751,
11757,
11769,
11787,
11793,
11829,
11841,
11901,
11967,
12003,
12009,
12021,
12039,
12057,
12063,
12081,
12099,
12147,
12153,
12171,
12219,
12237,
12273,
12279,
12297,
12333,
12381,
12387,
12399,
12417,
12459,
12471,
12477,
12531,
12603,
12633,
12651,
12657,
12687.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 173000 values, from 9 to 7043133).
n\r | 0 | 1 |
2 | 0 | 173000 | 2 |
3 | 172998 | 1 | 1 | 3 |
4 | 0 | 86459 | 0 | 86541 | 4 |
5 | 44 | 43222 | 43157 | 43311 | 43266 | 5 |
6 | 0 | 1 | 0 | 172998 | 0 | 1 | 6 |
7 | 87 | 28868 | 28838 | 28815 | 28825 | 28804 | 28763 | 7 |
8 | 0 | 43205 | 0 | 43262 | 0 | 43254 | 0 | 43279 | 8 |
9 | 63 | 1 | 0 | 86463 | 0 | 1 | 86472 | 0 | 0 | 9 |
10 | 0 | 43222 | 0 | 43311 | 0 | 44 | 0 | 43157 | 0 | 43266 | 10 |
11 | 34 | 17269 | 17317 | 17339 | 17281 | 17315 | 17277 | 17274 | 17287 | 17284 | 17323 |
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.