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aban n.  and  repunit

First numbers that belong to both sets:

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, 2000719, 2000811, 6000051, 8000413, 22000791, 23000461, 43000807, 54000453, 98000101, 168000483, 176000023, 185000803, 207000157, 244000021, 245000757, 259000743, 261000181, 317000221, 321000973, 377000473, 419000431, 470000721, 475000231, 482000071, 497000143, 527000893, 591000411, 617000761, 633000441, 656000157, 715000861, 732000081, 792000307, 810000061, 839000191, 874000533, 882000903, 916000491, 952000171, 1010000181, 1104000303, 1142000643, 1172000991, 1220000113, 1291000831, 1317000391, 1347000103, 1468000911, 1545000943, 1555000923, 1561000591, 1584000201, 1599000157, 1601000157, 1636000257, 1651000057, 1830000063, 1870000293, 1881000271, 1946000883, 2042000533, 2196000183, 2199000343, 2289000493, 2381000821, 2420000443, 2539000933, 2659000791, 2822000007, 2837000433, 2896000411, 2903000521, 2958000157, 3106000093, 3425000053, 3539000611, 3562000807, 3617000023, 3704000461, 3813000751, 3824000083, 3875000251, 4150000821, 4305000157, 4338000633, 4400000557, 4751000257, 4793000593, 5401000573, 5411000041, 5498000053, 6100000507, 6105000091, 6398000157, 6402000157, 6485000371, 6588000723, 6871000773, 7045000291, 7100000383, 7152000331, 7212000853, 7283000941, 7290000543, 7404000163, 8046000301, 8344000371, 8573000691, 8909000157, 9132000283, 9489000333, 9859000557, 9860000865, 10110000853, 10142000557, 10737000781, 10869000771, 11108000631, 11154000157, 11956000993, 11963000001, 12123000921, 13669000311, 13820000923, 14004000583, 14397000157, 14403000157, 14859000507, 15392000161, 16210000443, 16323000883, 16470000561, 16781000223, 17097000781, 17180000257, 17867000557, 17969000353, 18060000157, 18759000333, 19204000663, 19689000807, 19705000251, 19773000073, 19782000553, 20004000661, 20217000783, 20931000301, 21132000793, 21203000157, 21573000007, 22083000213, 22299000913, 22910000961, 23027000263, 23186000631, 23571000313, 24358000971, 24668000661, 25398000057, 25457000257, 25472000401, 25596000157, 25604000157, 25711000063, 26297000733, 26827000311, 26971000213, 27954000831, 28039000153, 28524000991, 28573000261, 28705000051, 29349000541, 29535000307, 29546000211, 29663000671, 30167000283, 30411000157, 30529000351, 30772000981, 30825000471, 31861000513, 32348000881, 32553000201, 33984000757, 34393000663, 34416000741, 34452000157, 34794000493, 34819000203, 35062000753, 35319000423, 35468000571, 35663000563, 35829000511, 36443000901, 36498000981, 37420000807, 37832000521, 39059000323, 39093000681, 39224000551, 39528000673, 39563000121, 39643000131, 39989000757, 39991000507, 39993000307, 39995000157, 39997000057, 39999000007, 40001000007, 40003000057, 40005000157, 40007000307, 40009000507, 40011000757, 40824000451, 41249000703, 41262000031, 41362000753, 41421000963, 41818000531, 42666000807, 43701000655, 43718000833, 44768000641, 45875000041, 45962000157, 46434000711, 46459000393, 46506000757, 46811000523, 47193000361.

Some larger examples :