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happy n.  and  powerful n.

First numbers that belong to both sets:

1, 32, 49, 100, 392, 784, 1000, 1125, 1152, 1323, 1444, 1521, 3200, 3364, 3456, 4096, 4225, 4356, 4563, 4900, 7225, 7744, 8281, 8748, 9604, 10000, 10125, 10976, 11025, 12167, 12321, 12348, 12996, 13824, 14112, 14283, 14641, 15625, 17424, 17672, 17956, 18432, 22201, 24025, 26136, 27225, 27556, 27889, 28561, 29584, 31212, 32000, 33800, 34596, 36504, 37636, 39200, 41472, 42025, 42592, 44217, 45369, 47432, 49000, 52272, 54289, 55696, 56644, 59049, 59536, 59643, 63504, 64009, 67712, 69192, 70225, 73441, 73984, 74088, 75625, 76729, 78400, 78408, 83521, 86528, 88804, 91592, 100000, 101124, 105625, 106929, 108241, 111392, 112500, 115200, 121032, 121104, 121203, 121500, 123201, 123823, 126025, 127449, 132300, 134456, 140625, 143641, 144400, 148176, 152100, 152352, 154568, 155236, 156025, 157609, 164025, 170569, 178929, 180625, 181476, 185761, 189728, 190969, 194688, 196249, 202612, 207025, 219488, 222264, 223112, 225792, 226981, 227529, 231125, 231525, 234256, 236196, 239121, 241081, 243675, 248004, 250047, 251001, 252004, 253125, 254016, 254043, 263169, 268324, 268912, 269361, 277729, 283024, 288463, 291848, 292681, 294849, 295936, 300763, 304175, 310472, 313632, 320000, 325125, 332929, 336400, 337561, 338000, 345600, 354025, 355216, 356409, 363312, 364500, 366368, 369800, 381024, 385875, 392000, 398161, 401408, 406125, 407044, 408321, 409600, 413449, 416025, 418176, 422500, 423801, 429025, 430336, 435483, 435600, 438012, 441800, 456300, 461041, 467856, 490000, 501264, 502681, 519841, 522729, 528529, 530604, 531441, 532512, 537824, 538756, 544500, 552123, 553352, 553536, 555579, 562432, 570025, 574564, 576583, 583443, 588289, 636192, 648675, 653400, 675684, 677448, 682276, 685584, 691488, 698896, 704969, 707281, 707472, 710432, 722500, 727609, 737881, 740772, 744012, 774400, 778688, 784000, 788768, 793016, 793117, 793800, 798848, 801025, 803736, 808201, 828100, 829939, 855468, 857476, 874800, 876488, 878004, 881792, 889249, 891136, 896809, 908209, 927472, 953312, 958441, 960400, 962361, 966289, 974408, 990125, 992016, 1000000, 1000188, 1010025, 1012500, 1013888, 1019592, 1025312, 1031048, 1032256.

Some larger examples :