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k-Lehmer numbers
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 
202103055 2 
310278208198310794 3 
40107113001031925 4 
51290451636385115058111672110895 5 
602081983010278010794 6 
72825601036864155029159771155379155542157910 7 
80539531051592405315990516001 8 
9072207436155095679118362151836807913558 9 
10016363850111672012904501150580110895 10 
11248283563510143684142576141943143488144075143591144453143232144220

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.