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happy n.  and  perfect power

First numbers that belong to both sets:

1, 32, 49, 100, 784, 1000, 1444, 1521, 3364, 4096, 4225, 4356, 4900, 7225, 7744, 8281, 9604, 10000, 11025, 12167, 12321, 12996, 13824, 14641, 15625, 17424, 17956, 22201, 24025, 27225, 27556, 27889, 28561, 29584, 34596, 37636, 42025, 45369, 54289, 55696, 56644, 59049, 59536, 63504, 64009, 70225, 73441, 73984, 74088, 75625, 76729, 78400, 83521, 88804, 100000, 101124, 105625, 106929, 108241, 121104, 123201, 126025, 127449, 140625, 143641, 144400, 152100, 155236, 156025, 157609, 164025, 170569, 178929, 180625, 181476, 185761, 190969, 196249, 207025, 226981, 227529, 234256, 236196, 239121, 241081, 248004, 250047, 251001, 252004, 254016, 263169, 268324, 269361, 277729, 283024, 292681, 294849, 295936, 300763, 332929, 336400, 337561, 354025, 355216, 356409, 398161, 407044, 408321, 409600, 413449, 416025, 422500, 423801, 429025, 430336, 435600, 461041, 467856, 490000, 501264, 502681, 519841, 522729, 528529, 531441, 537824, 538756, 553536, 570025, 574564, 588289, 675684, 682276, 685584, 698896, 704969, 707281, 722500, 727609, 737881, 774400, 778688, 801025, 808201, 828100, 857476, 889249, 891136, 896809, 908209, 958441, 960400, 962361, 966289, 992016, 1000000, 1010025, 1032256, 1044484, 1050625, 1054729, 1065024, 1092727, 1096209, 1100401, 1102500, 1106704, 1138489, 1177225, 1199025, 1205604, 1229881, 1232100, 1234321, 1240996, 1249924, 1267876, 1285956, 1290496, 1299600, 1304164, 1320201, 1329409, 1336336, 1364224, 1380625, 1385329, 1387684, 1394761, 1430416, 1444804, 1447209, 1456849, 1464100, 1468944, 1471369, 1481544, 1500625, 1507984, 1562500, 1575025, 1600225, 1633284, 1658944, 1682209, 1687401, 1700416, 1710864, 1734489, 1737124, 1739761, 1742400, 1774224, 1790244, 1795600, 1833316, 1836025, 1934881, 1948816, 1974025, 2007889, 2024929, 2085136, 2090916, 2093809, 2099601, 2108304, 2208196, 2220100, 2223081, 2226064, 2229049, 2334784, 2353156, 2393209, 2399401, 2402500, 2405601, 2411809, 2460375, 2477476, 2505889, 2518569, 2563201, 2566404, 2569609, 2579236, 2630884, 2637376, 2666689, 2715904, 2722500, 2755600, 2762244, 2772225, 2788900, 2803221, 2812329, 2819041, 2842596, 2856100, 2903616, 2920681, 2934369, 2948089, 2958400, 2961841, 2965284, 2968729, 2982529, 2989441, 2999824, 3003289, 3045025, 3055504, 3066001, 3069504, 3087049, 3090564, 3104644.

Some larger examples :