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D-numbers
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 
20173000 2 
317299811 3 
4086459086541 4 
54443222431574331143266 5 
601017299801 6 
787288682883828815288252880428763 7 
8043205043262043254043279 8 
9631086463018647200 9 
10043222043311044043157043266 10 
113417269173171733917281173151727717274172871728417323

A pictorial representation of the table above
motab
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.