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d-powerful n.  and  equidigital n.

First numbers that belong to both sets:

1, 2, 3, 5, 7, 43, 89, 135, 175, 224, 267, 283, 371, 373, 375, 379, 445, 463, 739, 1073, 1234, 1255, 1306, 1323, 1349, 1765, 2063, 2083, 2131, 2133, 2135, 2137, 2179, 2203, 2225, 2240, 2241, 2243, 2245, 2263, 2283, 2315, 2333, 2335, 2336, 2395, 2407, 2423, 2427, 2443, 2449, 2517, 2537, 2577, 2597, 2623, 2627, 2645, 2733, 2735, 2738, 2753, 2757, 2773, 2803, 2843, 2863, 2933, 2939, 2953, 2973, 3167, 3215, 3235, 3257, 3295, 3362, 3385, 3437, 3457, 3527, 3569, 3635, 3941, 3943, 3947, 4133, 4151, 4153, 4157, 4159, 4241, 4243, 4249, 4261, 4265, 4285, 4288, 4339, 4352, 4373, 4397, 4405, 4447, 4649, 4685, 4733, 4735, 4736, 4933, 4951, 4955, 4957, 5323, 5327, 5547, 5632, 5653, 5657, 5832, 5873, 6247, 6793, 6937, 7639, 8209, 8243, 8263, 8443, 8623, 8647, 8971, 8978, 9697, 9947, 9963, 10693, 10694, 10723, 12343, 12345, 12347, 12367, 12434, 12542, 12615, 12634, 12695, 12723, 12854, 13133, 13173, 13337, 13373, 13393, 13443, 13933, 14295, 14363, 14369, 14786, 15423, 16213, 16234, 16293, 16363, 16431, 16433, 16437, 16438, 16439, 16453, 16474, 16475, 16477, 16543, 16634, 16729, 16743, 16927, 17026, 17133, 17241, 17243, 17245, 17246, 17247, 17249, 17287, 17338, 17378, 17443, 17463, 17483, 17737, 17843, 17845, 17846, 17847, 18347, 18522, 18617, 18697, 18946, 19243, 19733, 19773, 20203, 20243, 20263, 20352, 20377, 20483, 20827, 20887, 20973, 21323, 21349, 21497, 21785, 22025, 22247, 22267, 22283, 22285, 22315, 22333, 22337, 22375, 22393, 22447, 22463, 22465, 22467, 22489, 22535, 22537, 22573, 22595, 22597, 22643, 22735, 22753, 22773, 22823, 22883, 22951, 22955, 22973, 23053, 23215, 23297, 23327, 23347, 23365, 23417, 23473, 23497, 23523, 23527, 23549, 23563, 23637, 23657, 23679, 23723, 23743, 23744, 23747, 23853, 23857, 23875, 23945, 23949, 23963, 24043, 24137, 24229, 24337, 24352.

Some larger examples :