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d-powerful n.  and  Smith n.

First numbers that belong to both sets:

4, 378, 1255, 2227, 2373, 2934, 3294, 3345, 3946, 4173, 4209, 4954, 4959, 6603, 6934, 8792, 8864, 9274, 15882, 16438, 16474, 17424, 17840, 18872, 22558, 22738, 23343, 24754, 24776, 25236, 25294, 26394, 27562, 27639, 28336, 29634, 32786, 32854, 32896, 32899, 32982, 32989, 33027, 33486, 33992, 34366, 34784, 34832, 34834, 34987, 35358, 35979, 36296, 36382, 36742, 36762, 36922, 37426, 37462, 37482, 38326, 39426, 39536, 39642, 43474, 43586, 43674, 43762, 46714, 46809, 47695, 47763, 49254, 51862, 52789, 52834, 53266, 53772, 57262, 58362, 59062, 59283, 59406, 59552, 59823, 59913, 59935, 62660, 63942, 64572, 64795, 66392, 67234, 67729, 68264, 68640, 69472, 69524, 71662, 73426, 73642, 73736, 74168, 74298, 74362, 75992, 76236, 76323, 76454, 76495, 78349, 79534, 79536, 79953, 82264, 82376, 82887, 83254, 83636, 83745, 84759, 84952, 85279, 85362, 85472, 87363, 91952, 92274, 92362, 92542, 92922, 92965, 94653, 96272, 97352, 98343, 98437, 123539, 124638, 125806, 125977, 126854, 131323, 132664, 132828, 132862, 132954, 137364, 137479, 137494, 137742, 137748, 138865, 139832, 142569, 142726, 145246, 145374, 146258, 146922, 147426, 149622, 152842, 153725, 153768, 156597, 158062, 162742, 164722, 166872, 167427, 175253, 177296, 177439, 179277, 182368, 184362, 184722, 184947, 186272, 192399, 192442, 192462, 193279, 194229, 195580, 196242, 205962, 209295, 209479, 222579, 222934, 223497, 223546, 223582, 224336, 224538, 224595, 224646, 224864, 224936, 225764, 225895, 226354, 226394, 226489, 226953, 229275, 229418, 229728, 229968, 231646, 232546, 232762, 232988, 234238, 234476, 234656, 235759, 236326, 236432, 236722, 236784, 237575, 237648, 237972, 238347, 239358, 239538, 239736, 239772, 239844, 239865, 242734, 243274, 243499, 243526, 243654, 243696, 243762, 243949, 244626, 244864, 245326, 245672, 245726, 246397, 248376, 248377, 248534, 248736, 248863, 249349, 249363, 249493, 251937, 252634, 252872, 253394, 253426, 254166, 254967, 256677, 256693, 257733, 257937, 258344, 259335, 259422, 262178, 262262, 262534, 262714, 262914, 263182, 263254, 263276, 263349, 263438, 263456.

Some larger examples :