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Harshad n.  and  modest n.

First numbers that belong to both sets:

111, 133, 209, 222, 266, 333, 399, 444, 511, 555, 666, 711, 777, 888, 999, 1011, 1015, 1066, 1090, 1233, 1236, 1422, 1545, 1854, 2022, 2030, 2090, 2133, 2466, 2844, 3033, 3055, 3654, 3699, 4044, 4060, 4108, 4148, 4266, 4872, 5055, 5066, 6066, 6090, 6111, 6399, 7077, 8088, 8148, 9099, 10011, 10090, 10101, 10185, 10270, 12036, 12108, 12222, 12324, 13111, 13117, 13351, 13481, 14126, 14259, 15045, 15111, 15135, 15405, 16074, 16144, 16296, 16432, 17153, 18054, 18162, 19171, 19513, 20009, 20022, 20030, 20108, 20202, 21189, 21567, 22198, 22594, 22814, 24111, 24444, 24648, 25675, 26027, 26222, 26333, 26481, 27037, 27729, 28063, 28756, 29783, 30033, 30222, 30303, 32144, 32148, 32296, 32592, 33111, 34074, 36054, 36108, 36162, 36324, 36666, 37999, 38114, 38703, 39081, 39333, 40012, 40044, 40060, 40404, 40740, 42111, 42999, 44814, 46074, 46111, 47141, 48216, 48222, 48888, 50055, 50303, 50370, 50505, 50925, 51111, 52074, 54162, 54486, 54729, 56666, 60066, 60090, 60111, 60135, 60180, 60270, 60405, 60444, 60540, 60606, 60810, 64144, 64288, 66222, 68102, 68148, 68333, 70011, 70074, 70077, 70101, 70210, 70518, 70707, 72108, 72162, 72216, 72324, 74222, 76114, 80024, 80036, 80088, 80370, 80592, 80808, 81729, 84126, 84222, 84252, 84777, 85185, 88132, 90099, 90101, 90162, 90270, 90666, 90810, 90909, 91333, 92138, 95555, 96144, 96216, 96222, 96432, 96444, 98518, 99333, 100011, 100090, 100101, 100110, 100165, 100180, 100185, 100270, 100300, 100303, 100450, 100540, 100740, 100900, 100909, 101010, 102222, 102459, 104148, 104216, 104962, 105111, 105135, 105315, 105405, 108162, 108324, 108444, 108486, 108729, 108972, 112216, 112333, 112336, 112518, 115185, 119259, 120022, 120033, 120036, 120099, 120108, 120132, 120180, 120216, 120222, 120303, 120324, 120360, 120540, 120666, 121212, 125454, 126162, 128296, 130117, 130234, 130351, 130702, 132444, 135729, 136296, 136306, 136444, 137411, 138222, 138621, 140022, 140126, 140148, 140202, 140252, 140259, 140370, 140630.

Some larger examples :