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Harshad n.  and  powerful n.

First numbers that belong to both sets:

1, 4, 8, 9, 27, 36, 72, 81, 100, 108, 144, 200, 216, 225, 243, 288, 324, 392, 400, 432, 441, 500, 512, 576, 648, 800, 864, 900, 972, 1000, 1125, 1152, 1296, 1323, 1521, 1728, 1764, 1800, 1944, 2000, 2025, 2304, 2312, 2401, 2592, 2601, 2700, 2704, 2916, 3200, 3456, 3528, 3600, 3888, 3969, 4000, 4225, 4356, 4500, 4608, 4624, 4913, 5000, 5184, 5292, 5400, 5832, 6084, 6400, 6912, 7056, 7200, 7744, 7776, 8000, 8100, 8575, 8712, 8748, 9000, 9216, 9747, 10000, 10125, 10368, 10404, 10584, 10800, 10816, 11025, 11664, 12100, 12168, 12321, 12348, 12544, 13068, 13500, 13689, 13824, 14112, 14400, 15552, 15876, 16200, 16875, 17424, 17496, 17576, 18000, 18252, 18432, 19044, 19600, 19683, 20000, 20736, 20808, 21168, 21600, 21904, 22500, 23328, 24200, 24300, 24336, 25947, 26136, 26244, 27000, 27648, 27783, 28224, 28800, 30276, 31104, 31212, 31752, 32000, 32400, 33124, 34300, 34992, 35152, 35937, 36000, 36100, 36125, 36504, 36963, 37044, 38988, 39200, 39204, 40000, 40401, 40500, 41472, 41616, 42336, 42592, 42849, 43200, 44100, 45000, 45387, 46656, 47628, 48400, 48600, 49923, 50000, 51200, 52272, 52488, 54000, 54675, 54756, 54925, 55296, 57132, 57600, 57800, 59049, 59319, 59643, 60500, 60516, 60552, 61504, 62001, 62208, 62424, 63504, 64000, 64800, 64827, 66248, 67500, 67712, 68600, 69696, 69984, 72000, 72900, 73008, 74088, 75625, 75843, 77841, 77976, 78408, 78732, 80000, 81000, 81675, 82944, 83232, 83349, 84375, 84672, 86400, 87723, 88200, 88209, 90000, 90828, 93312, 93636, 95256, 97200, 98568, 98784, 99225, 100000, 101124, 103041, 103788, 104329, 104544, 104976, 108000, 108900, 109512, 110592, 110889, 111132, 112500, 114264, 115200, 116964, 117612, 121000, 121032, 121104, 121203, 121500, 122500, 123201, 123823, 124416, 124848, 125000, 125316, 127008, 128547, 129600, 132300, 133128, 135000, 139876, 139968, 141512, 142884, 143748, 143883, 144000, 145800, 146016, 147852.

Some larger examples :