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d-powerful n.  and  super-d n.

First numbers that belong to both sets:

332, 333, 334, 739, 2131, 2373, 2395, 2532, 2627, 3324, 3342, 3345, 3362, 3385, 3569, 4261, 4269, 4285, 4469, 4649, 4732, 4869, 5342, 6669, 6792, 6793, 7632, 7836, 8623, 9963, 10693, 10694, 10722, 10723, 12495, 12695, 14369, 16213, 16431, 16471, 17848, 17914, 18347, 18617, 20203, 22269, 22315, 22335, 22373, 22538, 22663, 22848, 22954, 23239, 23297, 23476, 23529, 23582, 23812, 24336, 24487, 24489, 24599, 24620, 24627, 24669, 24681, 24682, 24712, 24716, 24733, 24735, 24736, 24737, 24739, 24750, 24751, 24752, 24753, 24754, 24755, 24756, 24757, 24758, 24759, 24773, 24776, 24792, 24796, 25127, 25431, 25749, 25769, 26736, 27369, 27498, 28269, 28731, 29343, 29346, 29347, 29349, 29369, 30347, 30348, 32072, 32781, 32810, 32818, 32819, 32831, 32907, 32969, 32981, 33220, 33221, 33222, 33223, 33224, 33225, 33226, 33227, 33228, 33229, 33242, 33246, 33247, 33260, 33261, 33262, 33263, 33264, 33265, 33266, 33267, 33268, 33269, 33283, 33284, 33286, 33288, 33318, 33338, 33372, 33398, 33422, 33427, 33428, 33429, 33442, 33447, 33467, 33486, 33512, 33532, 33538, 33539, 33578, 33592, 33622, 33623, 33624, 33648, 33685, 33686, 33752, 33757, 33772, 33775, 33778, 33779, 33792, 33798, 33804, 33822, 33823, 33824, 33826, 33828, 33829, 33842, 33843, 33844, 33846, 33847, 33864, 33865, 33882, 33884, 34231, 34295, 34498, 34652, 34873, 34981, 35081, 35248, 35427, 35931, 35932, 35939, 36457, 36892, 36981, 36988, 36989, 37002, 37267, 37645, 39469, 39624, 39682, 39869, 41727, 42263, 42269, 42519, 42536, 42687, 42689, 43434, 43762, 46715, 46719, 46731, 46863, 46931, 46932, 47690, 49720, 49891, 49893, 49894, 49895, 49896, 49926, 52432, 53396, 53469, 53480, 53481, 53483, 53488, 53489, 53862, 53956, 54369, 54873, 55226, 56859, 58362, 59069, 59193, 59242, 59269, 59339, 59931, 62139, 62394, 62469, 62519, 62539, 62642, 62669, 63294, 63471, 63492, 63493, 63766, 63769, 64624, 64731, 64733, 64739, 65672, 66481, 67232, 67743, 67898, 67904.

Some larger examples :