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d-powerful n.  and  Duffinian n.

First numbers that belong to both sets:

4, 8, 9, 63, 175, 209, 333, 371, 377, 407, 629, 849, 935, 1073, 1255, 1349, 1765, 2048, 2133, 2135, 2227, 2249, 2263, 2315, 2335, 2379, 2407, 2449, 2483, 2537, 2577, 2597, 2623, 2627, 2645, 2735, 2737, 2738, 2757, 2773, 2849, 2863, 2973, 3145, 3215, 3235, 3362, 3385, 3437, 3477, 3523, 3569, 3635, 3749, 3941, 3949, 4151, 4225, 4247, 4249, 4265, 4267, 4269, 4285, 4355, 4403, 4405, 4449, 4469, 4573, 4599, 4624, 4685, 4735, 4849, 4869, 4953, 4955, 5259, 5321, 5327, 5329, 5367, 5567, 5833, 6249, 6443, 6649, 6669, 6937, 7123, 7279, 7299, 7363, 8201, 8203, 8207, 8225, 8245, 8289, 8825, 8973, 8977, 8978, 9237, 9347, 9385, 9493, 9723, 9947, 10693, 12367, 12415, 13133, 13373, 13393, 14363, 16213, 16293, 16437, 16439, 16473, 16479, 16543, 16743, 16857, 17157, 17243, 17247, 17249, 17287, 17357, 17424, 17463, 17841, 17843, 17845, 17847, 17849, 18347, 18697, 19243, 19733, 20203, 20243, 20263, 20539, 20843, 20933, 20973, 21235, 21349, 21439, 21743, 21785, 21835, 22025, 22195, 22243, 22249, 22267, 22285, 22315, 22333, 22355, 22373, 22393, 22463, 22465, 22467, 22489, 22535, 22537, 22557, 22597, 22663, 22733, 22735, 22753, 22773, 22823, 22825, 22883, 22951, 22953, 22955, 22957, 22959, 23215, 23237, 23239, 23347, 23365, 23437, 23439, 23477, 23527, 23543, 23583, 23587, 23637, 23657, 23673, 23679, 23723, 23749, 23763, 23853, 23875, 23897, 23907, 23947, 23963, 24157, 24245, 24265, 24289, 24353, 24375, 24397, 24467, 24483, 24487, 24489, 24591, 24599, 24607, 24625, 24627, 24629, 24643, 24667, 24681, 24685, 24737, 24739, 24751, 24753, 24755, 24757, 24773, 24803, 24823, 24863, 24865, 24883, 24937, 25235, 25257, 25273, 25275, 25327, 25433, 25435, 25437, 25473, 25477, 25547, 25637, 25659, 25723, 25769, 26243, 26244, 26247, 26333, 26337, 26443, 26483, 26533, 26555, 26593, 26739, 26807, 27129, 27237, 27323, 27347, 27349, 27363.

Some larger examples :