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cyclic n.  and  Harshad n.

First numbers that belong to both sets:

1, 2, 3, 5, 7, 133, 209, 247, 407, 481, 511, 629, 803, 1141, 1387, 1417, 1547, 1651, 1679, 1729, 1853, 2001, 2119, 2159, 2353, 2465, 2755, 2771, 2821, 3031, 3055, 3077, 3097, 3383, 3439, 3523, 3749, 3781, 4465, 4577, 4607, 4991, 5149, 5161, 5833, 6331, 6443, 7033, 7055, 7361, 7501, 7543, 7859, 8203, 8227, 8717, 8911, 9253, 9503, 9545, 10011, 10109, 10213, 10279, 10307, 10543, 10703, 10727, 10787, 10963, 11245, 11339, 11647, 11951, 12563, 12673, 12857, 13111, 13117, 13481, 13515, 13685, 14093, 14099, 14383, 14521, 14705, 15045, 15067, 15409, 15689, 15751, 15755, 16169, 16435, 16541, 16583, 17119, 17153, 17461, 17777, 17803, 17887, 18031, 18071, 18239, 18299, 18653, 19171, 19601, 19865, 20001, 20009, 20137, 20197, 20355, 20371, 20445, 20603, 20605, 20801, 20881, 21073, 21165, 21301, 21307, 21437, 21541, 21565, 21743, 21907, 22009, 22049, 22249, 22355, 22591, 22661, 22711, 22793, 22933, 22997, 23055, 23413, 23617, 24191, 24449, 24643, 24803, 25051, 25277, 25327, 25415, 25691, 25699, 26221, 26333, 26519, 26933, 27037, 27251, 27347, 27721, 27761, 28405, 28489, 28739, 29417, 29431, 29831, 30121, 30503, 30617, 30659, 30701, 30705, 30901, 31135, 31229, 31483, 31535, 31841, 32011, 32071, 32147, 32167, 32453, 32729, 33007, 33065, 33193, 33241, 33371, 33535, 33557, 33959, 34219, 34411, 34561, 34901, 34903, 35113, 35207, 35245, 35513, 35627, 36105, 36271, 36455, 36569, 36613, 36859, 37043, 37283, 38323, 38657, 39649, 40007, 40027, 40201, 40261, 40403, 40409, 40595, 40601, 40715, 40717, 41059, 41431, 41633, 41789, 41837, 42085, 42551, 42601, 42665, 42997, 43079, 43163, 43303, 43453, 43493, 43877, 44081, 44115, 44137, 44399, 44735, 44979, 45149, 45163, 45505, 45563, 45611, 45965, 46111, 46223, 46391, 46531, 47141, 47219, 47633, 47989, 48047, 48241, 48577, 49097, 49693, 49703, 49987, 50235, 50293, 50303, 50501, 50507, 50945, 51119, 51319, 51405, 51661, 51773.

Some larger examples :