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d-powerful n.  and  palindromic n.

First numbers that belong to both sets:

1, 2, 3, 4, 5, 6, 7, 8, 9, 262, 333, 373, 2332, 4114, 4224, 8228, 23232, 24842, 27472, 32923, 38383, 39393, 42624, 43434, 44644, 49894, 52325, 52725, 59795, 63436, 63836, 73537, 79597, 98389, 236632, 237732, 243342, 249942, 262262, 264462, 265562, 283382, 435534, 473374, 623326, 635536, 638836, 693396, 826628, 827728, 837738, 873378, 1679761, 2134312, 2138312, 2230322, 2234322, 2236322, 2238322, 2246422, 2252522, 2254522, 2256522, 2258522, 2294922, 2296922, 2298922, 2322232, 2334332, 2336332, 2340432, 2344432, 2346432, 2350532, 2352532, 2354532, 2356532, 2358532, 2378732, 2380832, 2382832, 2384832, 2386832, 2426242, 2436342, 2438342, 2458542, 2472742, 2524252, 2526252, 2532352, 2534352, 2536352, 2570752, 2572752, 2574752, 2592952, 2594952, 2632362, 2634362, 2638362, 2648462, 2652562, 2658562, 2668662, 2670762, 2694962, 2696962, 2736372, 2740472, 2744472, 2762672, 2764672, 2796972, 2832382, 2834382, 2836382, 2894982, 2932392, 2974792, 3223223, 3225223, 3245423, 3249423, 3253523, 3269623, 3273723, 3277723, 3307033, 3323233, 3345433, 3363633, 3423243, 3425243, 3427243, 3431343, 3437343, 3467643, 3471743, 3473743, 3479743, 3497943, 3527253, 3549453, 3551553, 3573753, 3583853, 3625263, 3629263, 3633363, 3655563, 3673763, 3725273, 3739373, 3765673, 3825283, 4220224, 4222224, 4228224, 4230324, 4232324, 4234324, 4236324, 4238324, 4262624, 4264624, 4268624, 4276724, 4324234, 4326234, 4328234, 4332334, 4342434, 4368634, 4372734, 4374734, 4376734, 4378734, 4392934, 4456544, 4462644, 4464644, 4466644, 4476744, 4522254, 4528254, 4546454, 4574754, 4576754, 4634364, 4642464, 4722274, 4732374, 4734374, 4736374, 4792974, 4832384, 4834384, 4922294, 4924294, 4928294, 4956594, 4962694, 4972794, 4976794, 5233325, 5235325, 5237325, 5239325, 5243425, 5253525, 5263625, 5277725, 5279725, 5297925, 5299925, 5323235, 5325235, 5327235, 5339335, 5347435, 5387835, 5395935, 5397935, 5423245, 5523255, 5543455, 5627265, 5781875, 5823285, 5927295, 6254526, 6292926, 6294926, 6324236, 6342436, 6372736, 6392936, 6426246, 6434346, 6494946, 6496946, 6554556, 6594956, 6724276, 6852586, 6854586, 6896986, 6972796, 6974796, 6994996, 6996996, 7223227, 7253527, 7255527, 7327237, 7349437, 7363637, 7365637, 7391937, 7423247, 7427247, 7433347, 7463647, 7465647, 7467647, 7521257, 7523257, 7565657, 7569657, 7623267, 7629267, 7637367, 7639367, 7643467, 7679767, 7697967, 7735377, 7785877, 7863687.

Some larger examples :