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Friedman n.  and  powerful n.

First numbers that belong to both sets:

25, 121, 125, 128, 216, 289, 343, 625, 1024, 1296, 2048, 2187, 2500, 2592, 2916, 3125, 3375, 4096, 4624, 5776, 5832, 8192, 9025, 9216, 9261, 10201, 10368, 11025, 11664, 12100, 12167, 12321, 12544, 12769, 12996, 13225, 13689, 13824, 14641, 15125, 15552, 15625, 16384, 16807, 16875, 17689, 18225, 18432, 19321, 19683, 19773, 20736, 21952, 23328, 24336, 24389, 24964, 25921, 26244, 26896, 27436, 27648, 27783, 27889, 28224, 28561, 28900, 29584, 29768, 29929, 30625, 32761, 32768, 34992, 35152, 35721, 35937, 36864, 37249, 38416, 39304, 41472, 42025, 42336, 42875, 43264, 45369, 46656, 48672, 51200, 52488, 53824, 54872, 55225, 55296, 59049, 59319, 65536, 67228, 68644, 69984, 70225, 73984, 76832, 78125, 78732, 81225, 82944, 83521, 84375, 84672, 85264, 91125, 93312, 97336, 99225, 103684, 103823, 103968, 104329, 104976, 105625, 106929, 107163, 107584, 107653, 108241, 111132, 112896, 114244, 114921, 117128, 117649, 118336, 119025, 121203, 123904, 124416, 124609, 125000, 125316, 126025, 126736, 127449, 128164, 129600, 130321, 131072, 131769, 132496, 132651, 133225, 133956, 134456, 134689, 135424, 136161, 137288, 137641, 139129, 139392, 139968, 140608, 140625, 141376, 142129, 142884, 143648, 143748, 145924, 146689, 147456, 148176, 148225, 151263, 151875, 152352, 153664, 157464, 159201, 161051, 162409, 163216, 164025, 165375, 165888, 166375, 167281, 169744, 170569, 171396, 172872, 175232, 175616, 176400, 177147, 177241, 178929, 180625, 181476, 183184, 184832, 185193, 186624, 187272, 188356, 189225, 191664, 192721, 194481, 194672, 195112, 195364, 196249, 198025, 202248, 204800, 205379, 207936, 209764, 209952, 210125, 210681, 214369, 214375, 215168, 215296, 216000, 216225, 217156, 218700, 219024, 219961, 221184, 221841, 222784, 223587, 223729, 224676, 224939, 225625, 225792, 226576, 226981, 227529, 228484, 228488, 228528, 231361, 231525, 232324, 233289, 233523, 233928, 234256, 235224, 235225, 236196, 236672, 237169, 237276, 238144, 238328, 238572, 239121, 239432, 241081, 242064, 243049, 243675, 244036, 247009, 248832, 249696, 250000, 250047, 251001, 252004, 253009, 253125.

Some larger examples :