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d-powerful n.  and  untouchable n.

First numbers that belong to both sets:

2, 5, 262, 372, 518, 1074, 1326, 1542, 2048, 2062, 2136, 2248, 2356, 2376, 2464, 2572, 2628, 2824, 2932, 2994, 3234, 3236, 3238, 3292, 3432, 3454, 3476, 3652, 3746, 3786, 3940, 3942, 4224, 4248, 4268, 4316, 4396, 4574, 4648, 4738, 4956, 5320, 5322, 5328, 5364, 5366, 5766, 5832, 7032, 7432, 7582, 7632, 7944, 7962, 8202, 8204, 8206, 8228, 8262, 8316, 8352, 8792, 8978, 9234, 9236, 9362, 10722, 12298, 12432, 12562, 14362, 15422, 16470, 17026, 17244, 17246, 17338, 17372, 17424, 17844, 17846, 17994, 19732, 20332, 20352, 20532, 22332, 22358, 22376, 22536, 22538, 22628, 22848, 22932, 22952, 23034, 23232, 23348, 23364, 23436, 23472, 23492, 23522, 23812, 23892, 23924, 23928, 24336, 24352, 24464, 24576, 24598, 24620, 24628, 24644, 24756, 24758, 24776, 24792, 24996, 25272, 25366, 25430, 25436, 25476, 25672, 25872, 26246, 26352, 27326, 27472, 27562, 28260, 28266, 28332, 28336, 28532, 28554, 28736, 28738, 28932, 30346, 30964, 31558, 32256, 32436, 32528, 32788, 32814, 32898, 32980, 33022, 33042, 33222, 33246, 33262, 33264, 33266, 33372, 33486, 33686, 33822, 33882, 33974, 33992, 34230, 34272, 34290, 34326, 34342, 34388, 34526, 34542, 34562, 34568, 34746, 34832, 34928, 34980, 34986, 34988, 35022, 35080, 35208, 35220, 35248, 35352, 35552, 35688, 35738, 35992, 36104, 36292, 36324, 36348, 36382, 36412, 36522, 36568, 36708, 36854, 36984, 37222, 37228, 37264, 37284, 37332, 37336, 37352, 37392, 37422, 37462, 37578, 37732, 37826, 38272, 38762, 39338, 39428, 39642, 39648, 39758, 41218, 42048, 42248, 42262, 42264, 42624, 42662, 43148, 43236, 43438, 43478, 43498, 43528, 43586, 43674, 44572, 44640, 44646, 44662, 44932, 46372, 46710, 46718, 46736, 46738, 46792, 46794, 46824, 46974, 46976, 47236, 47256, 47346, 47690, 47694, 47696, 47928, 47946, 47962, 48936, 49168, 49236, 49238, 49276, 49542, 49678, 49720, 49724, 49762, 49836, 49896, 49898, 49968, 52344, 52432, 52702, 52704, 52708.

Some larger examples :