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amenable n.  and  hyperperfect n.

First numbers that belong to both sets:

21, 28, 301, 325, 496, 697, 1333, 1909, 2041, 2133, 3901, 8128, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1284121, 1403221, 1433701, 1570153, 1618597, 1787917, 1950625, 2287933, 2462881, 2469601, 2488201, 2666437, 2708413, 2768581, 2856481, 2884501, 2924101, 3328921, 3420301, 4013833, 4304341, 4312681, 4658449, 5199013, 6392257, 7152001, 7478041, 8766061, 8883841, 9398593, 9427657, 9699181, 10445221, 12064333, 12160249, 12283693, 13646989, 14444041, 14489437, 15042553, 15407173, 16260901, 16641241, 16904101, 18116737, 19106449, 20103001, 20151517, 20419093, 21396313, 23548753, 25773901, 26249473, 26585233, 28005301, 28449901, 28600321, 29751229, 29757601, 30374941, 31696501, 31837801, 32545741, 33550336, 34251601, 35315221, 35492533, 35494981, 36171853, 36640993, 37021141, 38159521, 38749153, 39116221, 39147301, 40374013, 41354041, 41881201, 42387097, 46024681, 46596457, 47014801, 47842237, 48373597, 50809261, 52606861, 53120149, 54296677, 55262737, 56979661, 59675773, 60110701, 61442077, 61599553, 62722153, 62804941, 63519481, 64220161, 65618101, 77939221, 79001833, 79089541, 81329701, 81705601, 88347781, 91736353, 94126657, 94472041, 95235601, 95295817, 97438381, 97585249, 101402761, 105070201, 106875541, 108240841, 112803841, 112891993, 113513761, 115788961, 116128021, 118535929, 119313541, 124035913, 125150281, 125438101, 127180033, 127802533, 128990941, 129127041, 132123973, 133815229, 134281141, 134927281, 134959501, 138167929, 138786229, 140520901, 147816121, 150366841, 151547761, 153346261, 155367733, 156796501, 157026301, 157767481, 158048281, 158813737, 160123261, 162157789, 163331269, 168159889, 170023489, 176344741, 179920057, 181132801, 181728817, 185560633, 186898501, 190986001, 191535121, 191847181, 192317941, 194283181, 196581361, 199839457, 204349393, 205100017, 205253533, 214696441, 217033693, 219575101, 221814409, 228382837, 228695497, 229971241, 234784201, 243000001, 243328381, 244930669, 247005973, 247941649, 255286081, 258051649, 264521701, 265434901, 267639481, 273239821, 275797861, 281517121, 289054369, 291235093, 291461701, 302343133, 312210169, 313255213, 318561193, 333080509, 346548133, 350992501, 353944177, 357858001, 359269021, 362320489, 375014317, 383550289, 383712781, 391854937, 395421517, 398383441, 401039341, 404458477, 407721793, 414857713, 417439801, 419745517, 420562573, 421569877, 430920001, 432373033, 434307901, 438281281, 440179273, 441116281, 445057057, 447422401, 448644961, 454166017, 454503817, 470912293, 476610241, 483805033, 495196453, 504662701, 509031157, 511017757, 512459221, 517093141, 528316933, 528793501, 529563217, 530710561.

Some larger examples :