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d-powerful n.  and  de Polignac n.

First numbers that belong to both sets:

1, 373, 2203, 2263, 2465, 2843, 3215, 4151, 4153, 4573, 4649, 5323, 6247, 7343, 13393, 16213, 16543, 17849, 20827, 22663, 22957, 23417, 23437, 23857, 24467, 24599, 24607, 24683, 24733, 24847, 24977, 25273, 25453, 25547, 25723, 26647, 26807, 27253, 27343, 27479, 28267, 28269, 29369, 30341, 32527, 32983, 33221, 33685, 34297, 34525, 35221, 35447, 36457, 37243, 37483, 37663, 39393, 39869, 42599, 42623, 45583, 45899, 46971, 46993, 47363, 47699, 49721, 49897, 52783, 54329, 54727, 55483, 59063, 59203, 59935, 60397, 60409, 62395, 62429, 62519, 62609, 62849, 63291, 63299, 64573, 65473, 65743, 65879, 66913, 67327, 68669, 69293, 69307, 72389, 73649, 74293, 74323, 74363, 75937, 76943, 77849, 78523, 81293, 82447, 83257, 83543, 85325, 89497, 92303, 93869, 98345, 98503, 98893, 117701, 119639, 119747, 123283, 123537, 123539, 124237, 124387, 124679, 124963, 125579, 125977, 126389, 126457, 132463, 132629, 134287, 134353, 134429, 135277, 137237, 137549, 137633, 137743, 139259, 139453, 141353, 142679, 142697, 142963, 143753, 143915, 145931, 148213, 156517, 157327, 160475, 164347, 165223, 172467, 172537, 177293, 177329, 177493, 177989, 178243, 178739, 179233, 179347, 179743, 183733, 183997, 184237, 184363, 190427, 192733, 195581, 196397, 196537, 202739, 204449, 205363, 205783, 205835, 209437, 209471, 212233, 213427, 213733, 214349, 214363, 214769, 215227, 215793, 215939, 222373, 222959, 223633, 223655, 223787, 223949, 223987, 224603, 225763, 226735, 227323, 227453, 227947, 228995, 229343, 229543, 230743, 231337, 232439, 232787, 233669, 235267, 235483, 235777, 235957, 236219, 236279, 236369, 236497, 236749, 236879, 236947, 237133, 237427, 237643, 237667, 237953, 237997, 238259, 238639, 238903, 239317, 239357, 239377, 239429, 239577, 240333, 241637, 242377, 243037, 243239, 243257, 243293, 243341, 243349, 243389, 243563, 243749, 243877, 243941, 244393, 244771, 245813, 246371, 246397, 248531, 248593, 248759, 248777, 249239, 249329, 249475, 249477, 249637, 249899, 250273, 252477, 252749, 252767, 253087, 253355, 253537, 253739, 254233, 254347, 255313, 255733, 255823, 256453, 256769, 256853, 256873.

Some larger examples :