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balanced prime  and  d-powerful n.

First numbers that belong to both sets:

5, 373, 17483, 22447, 23327, 26393, 42683, 55843, 60337, 63493, 63527, 63697, 98473, 126839, 164279, 179743, 213223, 226643, 236707, 239423, 243233, 243367, 249329, 253537, 267587, 267893, 273629, 284539, 288577, 297607, 304943, 327667, 328277, 340903, 341623, 342653, 344257, 346349, 347759, 348253, 358607, 398267, 422557, 427523, 432743, 441613, 443237, 443413, 445877, 452233, 456439, 463747, 496493, 525247, 527063, 527447, 528673, 529933, 532193, 533453, 536287, 538823, 567667, 625637, 626953, 634273, 642247, 642487, 655267, 656297, 657269, 696497, 729643, 788363, 794537, 820643, 823547, 845723, 862997, 922457, 1032751, 1033783, 1049827, 1049843, 1063649, 1081429, 1169347, 1226347, 1232339, 1236797, 1244567, 1253723, 1294627, 1376497, 1428893, 1429843, 1437257, 1473743, 1487429, 1597753, 1619747, 1643357, 1644299, 1653433, 1654733, 1663867, 1674643, 1682567, 1688237, 1696517, 1698247, 1724357, 1724683, 1727969, 1729493, 1753883, 1760953, 1775629, 1830623, 1857893, 1935287, 1953157, 1953863, 1956743, 1958563, 1959827, 1964623, 1966337, 1975663, 1976543, 2005453, 2015359, 2065633, 2066957, 2079047, 2085569, 2085983, 2092543, 2125933, 2126633, 2132551, 2134267, 2162929, 2167339, 2177437, 2188987, 2209693, 2215393, 2219557, 2222573, 2223037, 2225653, 2228687, 2233207, 2235803, 2243497, 2245637, 2246371, 2251643, 2253827, 2254507, 2255263, 2255399, 2257397, 2257867, 2263369, 2263433, 2264803, 2266337, 2268443, 2275937, 2276059, 2278453, 2278637, 2284957, 2285273, 2285323, 2286623, 2287387, 2289767, 2293579, 2293919, 2322253, 2322367, 2322967, 2324873, 2325773, 2327069, 2327539, 2333657, 2337547, 2340763, 2347627, 2353411, 2358463, 2362147, 2365283, 2367823, 2372737, 2383643, 2384653, 2385953, 2387243, 2389427, 2393653, 2394157, 2398267, 2399273, 2418439, 2423789, 2424553, 2426131, 2429723, 2430563, 2432257, 2434763, 2437577, 2439953, 2441723, 2452903, 2453827, 2454343, 2458327, 2458837, 2462287, 2468357, 2469653, 2470673, 2476297, 2478367, 2480551, 2485453, 2488553, 2492453, 2497667, 2498213, 2504323, 2505533, 2506723, 2514263, 2518643, 2518933, 2520257, 2524673, 2526473, 2526547, 2527397, 2528657, 2534443, 2539343, 2544637, 2566673, 2574757, 2575123, 2575493, 2575543, 2584943, 2588567, 2599297, 2631043, 2632543, 2635331, 2635429, 2635973, 2643307, 2644823, 2648449, 2664247, 2667053, 2668577, 2669237, 2673283, 2679973, 2684599, 2687123, 2689543, 2698499, 2704963, 2729423, 2739637.

Some larger examples :