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apocalyptic n.  and  d-powerful n.

First numbers that belong to both sets:

224, 226, 994, 1349, 1634, 2064, 2083, 2114, 2130, 2131, 2132, 2133, 2134, 2138, 2223, 2224, 2225, 2227, 2240, 2242, 2243, 2245, 2246, 2247, 2283, 2315, 2333, 2335, 2352, 2353, 2356, 2372, 2395, 2396, 2407, 2443, 2445, 2449, 2464, 2532, 2537, 2572, 2574, 2607, 2626, 2627, 2628, 2733, 2735, 2738, 2739, 2757, 2792, 2793, 2794, 2824, 2843, 2849, 2863, 2864, 2932, 2933, 2934, 2939, 2953, 2973, 2994, 3165, 3215, 3236, 3238, 3254, 3257, 3274, 3294, 3295, 3298, 3324, 3342, 3385, 3432, 3454, 3492, 3523, 3525, 3566, 3569, 3612, 3652, 3657, 3678, 3692, 3706, 3725, 3749, 3765, 3786, 3856, 3940, 3941, 3942, 3943, 3944, 3945, 3946, 3947, 3948, 4132, 4136, 4150, 4151, 4152, 4155, 4156, 4157, 4159, 4194, 4209, 4225, 4240, 4242, 4244, 4245, 4246, 4247, 4249, 4260, 4261, 4263, 4264, 4265, 4267, 4269, 4285, 4316, 4339, 4352, 4355, 4396, 4405, 4424, 4443, 4447, 4449, 4572, 4573, 4574, 4599, 4622, 4624, 4644, 4649, 4685, 4733, 4735, 4736, 4738, 4826, 4849, 4869, 4933, 4951, 4952, 4953, 4954, 4955, 4957, 4958, 5182, 5258, 5259, 5276, 5321, 5322, 5323, 5324, 5325, 5326, 5327, 5328, 5329, 5342, 5344, 5364, 5366, 5367, 5384, 5453, 5547, 5562, 5653, 5657, 5746, 5832, 5858, 5877, 6244, 6249, 6424, 6443, 6579, 6603, 6649, 6714, 6732, 6792, 6793, 6794, 6849, 6932, 6934, 6935, 6937, 6939, 6972, 6974, 6976, 7032, 7123, 7234, 7279, 7296, 7299, 7329, 7432, 7632, 7639, 7836, 7906, 7962, 7964, 8200, 8201, 8202, 8203, 8204, 8205, 8207, 8208, 8209, 8223, 8224, 8225, 8243, 8245, 8262, 8263, 8283, 8288, 8289, 8316, 8352, 8376, 8443, 8536, 8623, 8647, 8667, 8712, 8756, 8792, 8825, 8864, 8970, 8971, 8972, 8973, 8974, 8975, 8977, 8978, 8979, 9122, 9234, 9237, 9272, 9273, 9274, 9324, 9342.

Some larger examples :