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repunits
A number whose digits in a given base are all equal to 1. more

The first 600 repunits :
7, 13, 15, 21, 31, 40, 43, 57, 63, 73, 85, 91, 111, 121, 127, 133, 156, 157, 183, 211, 241, 255, 259, 273, 307, 341, 343, 364, 381, 400, 421, 463, 507, 511, 553, 585, 601, 651, 703, 757, 781, 813, 820, 871, 931, 993, 1023, 1057, 1093, 1111, 1123, 1191, 1261, 1333, 1365, 1407, 1464, 1483, 1555, 1561, 1641, 1723, 1807, 1885, 1893, 1981, 2047, 2071, 2163, 2257, 2353, 2380, 2451, 2551, 2653, 2757, 2801, 2863, 2955, 2971, 3081, 3193, 3280, 3307, 3423, 3541, 3616, 3661, 3783, 3906, 3907, 4033, 4095, 4161, 4291, 4369, 4423, 4557, 4681, 4693, 4831, 4971, 5113, 5220, 5257, 5403, 5461, 5551, 5701, 5853, 6007, 6163, 6175, 6321, 6481, 6643, 6807, 6973, 7141, 7240, 7311, 7381, 7483, 7657, 7833, 8011, 8191, 8373, 8421, 8557, 8743, 8931, 9121, 9313, 9331, 9507, 9703, 9724, 9841, 9901, 10101, 10303, 10507, 10713, 10921, 11111, 11131, 11155, 11343, 11557, 11773, 11991, 12211, 12433, 12657, 12720, 12883, 13111, 13341, 13573, 13807, 14043, 14281, 14425, 14521, 14763, 15007, 15253, 15501, 15751, 16003, 16105, 16257, 16276, 16383, 16513, 16771, 17031, 17293, 17557, 17823, 18091, 18279, 18361, 18633, 18907, 19183, 19461, 19531, 19608, 19741, 20023, 20307, 20440, 20593, 20881, 21171, 21463, 21757, 21845, 22053, 22351, 22621, 22651, 22765, 22953, 23257, 23563, 23871, 24181, 24493, 24807, 25123, 25260, 25441, 25761, 26083, 26407, 26733, 27061, 27391, 27723, 27931, 28057, 28393, 28731, 29071, 29413, 29524, 29757, 30103, 30451, 30784, 30801, 30941, 31153, 31507, 31863, 32221, 32581, 32767, 32943, 33307, 33673, 33825, 34041, 34411, 34783, 35157, 35533, 35911, 36291, 36673, 37057, 37060, 37443, 37449, 37831, 38221, 38613, 39007, 39403, 39801, 40201, 40495, 40603, 41007, 41371, 41413, 41821, 42231, 42643, 43057, 43473, 43891, 44136, 44311, 44733, 45157, 45583, 46011, 46441, 46873, 47307, 47743, 47989, 48181, 48621, 49063, 49507, 49953, 50401, 50851, 51303, 51757, 52060, 52213, 52671, 53131, 53593, 54057, 54241, 54523, 54991, 55461, 55933, 55987, 56355, 56407, 56883, 57361, 57841, 58323, 58807, 59293, 59781, 60271, 60763, 60880, 61257, 61753, 62251, 62751, 63253, 63757, 64263, 64771, 65281, 65535, 65641, 65793, 66307, 66430, 66823, 67341, 67861, 68383, 68907, 69433, 69905, 69961, 70491, 70644, 71023, 71557, 72093, 72631, 73171, 73713, 74257, 74803, 75351, 75895, 75901, 76453, 77007, 77563, 78121, 78681, 79243, 79807, 80373, 80941, 81400, 81511, 82083, 82657, 83233, 83811, 84391, 84973, 85557, 86143, 86731, 87165, 87321, 87381, 87913, 88507, 88573, 88741, 89103, 89701, 90301, 90903, 91507, 92113, 92721, 93196, 93331, 93943, 94557, 95173, 95791, 96411, 97033, 97656, 97657, 98283, 98911, 99499, 99541, 100173, 100807, 101443, 102081, 102721, 103363, 104007, 104653, 105301, 105951, 106080, 106603, 107257, 107913, 108571, 109231, 109893, 110557, 111111, 111151, 111223, 111891, 112561, 112945, 113233, 113907, 114583, 115261, 115941, 116623, 117307, 117993, 118681, 119371, 120063, 120100, 120757, 121453, 122151, 122851, 123553, 124257, 124963, 125671, 126381, 127093, 127551, 127807, 128523, 129241, 129961, 130683, 131071, 131407, 132133, 132861, 133591, 134323, 135057, 135304, 135793, 136531, 137257, 137271, 137561, 138013, 138757, 139503, 140251, 141001, 141753, 142507, 143263, 143365, 144021, 144781, 145543, 146307, 147073, 147841, 148611, 149383, 150157, 150933, 151711, 151740, 152491, 153273, 154057, 154843, 155631, 156421, 157213, 158007, 158803, 159601, 160401, 160435, 161203, 162007, 162813, 163621, 164431, 165243, 166057, 166873, 167691, 168421, 168511, 169333, 169456, 170157, 170983, 171811, 172641, 173473, 174307, 175143, 175981, 176821, 177156, 177663, 178507, 178809, 179353, 180201, 181051, 181903, 182757, 183613, 184471, 185331, 186193, 187057, 187923, 188500, 188791, 189661, 190533, 191407, 192283, 193161, 194041, 194923, 195807, 196693, 197581, 198471, 198535, 199363, 200257, 201153, 202051, 202951, 203853, 204205, 204757, 205663, 206571, 207481, 208393, 208920, 209307, 210223, 211141, 212061, 212983, 213907, 214833, 215761, 216691, 217623, 218557, 219493, 219661, 220431, 221371, 222313, 223257, 224203, 225151, 226101, 227053, 228007, 228963, 229921, 230764, 230881, 231843, 232807, 233773, 234741, 235711, 236683, 237657, 238633, 239611, 240591.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 31730131 values, from 7 to 999999993568953).

n\r 0  1 
25062131679510 2 
310575057211531371937 3 
4503561584117226515838338 4 
5614671267429363247021264943720232 5 
617042211195822410558015335551913 6 
79050214908008615115903593815122159344517722 7 
825214792059613379191522514279205761327919186 8 
9115077039970645105520747073206646114767039961646 9 
103016412664068212649318101113130310225632470011910121 10 
111129257602935767825574971327832674189505759246927028753455749691

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.