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frugal n.  and  square n.

First numbers that belong to both sets:

256, 625, 729, 1024, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10000, 10201, 10609, 11449, 11664, 11881, 12544, 12769, 14641, 15625, 16129, 16384, 17161, 18225, 18769, 19321, 20736, 21609, 22201, 22801, 24649, 26244, 26569, 27889, 28561, 29929, 30625, 32041, 32761, 35721, 36481, 37249, 38416, 38809, 39601, 40000, 44521, 46656, 49729, 50625, 51529, 52441, 54289, 57121, 58081, 59049, 60025, 62500, 63001, 65536, 66049, 69169, 72361, 73441, 76729, 78961, 80089, 83521, 85849, 94249, 96721, 97969, 100489, 102400, 104976, 105625, 107584, 108241, 109561, 110224, 110889, 112225, 113569, 114244, 117649, 118336, 120409, 121801, 123201, 124609, 126025, 126736, 128881, 130321, 131769, 133225, 134689, 135424, 136161, 137641, 139129, 140625, 141376, 143641, 146689, 147456, 149769, 150544, 151321, 153664, 156025, 157609, 160000, 160801, 164025, 167281, 170569, 172225, 175561, 177241, 178929, 179776, 180625, 182329, 185761, 186624, 187489, 192721, 194481, 196249, 198025, 200704, 201601, 208849, 210681, 212521, 214369, 215296, 218089, 219961, 222784, 225625, 227529, 229441, 234256, 235225, 236196, 237169, 238144, 241081, 246016, 247009, 249001, 250000, 253009, 257049, 259081, 261121, 262144, 263169, 271441, 273529, 279841, 281961, 287296, 290521, 292681, 299209, 301401, 305809, 310249, 316969, 321489, 322624, 323761, 326041, 330625, 331776, 332929, 334084, 337561, 341056, 344569, 350464, 351649, 358801, 361201, 363609, 366025, 368449, 375769, 380689, 383161, 385641, 388129, 390625, 398161, 399424, 405769, 408321, 409600, 410881, 413449, 418609, 419904, 426409, 430336, 431649, 434281, 436921, 440896, 452929, 455625, 456976, 458329, 461041, 466489, 470596, 473344, 477481, 491401, 502681, 505521, 506944, 516961, 521284, 525625, 528529, 531441, 537289, 546121, 552049, 558009, 564001, 565504, 573049, 579121, 589824, 591361, 597529, 600625, 602176, 613089, 614656, 619369, 635209, 640000, 641601, 654481, 657721, 674041, 677329, 683929, 687241, 693889, 700569, 703921, 707281, 714025, 717409, 719104, 727609, 734449, 737881, 744769, 746496, 751689, 762129, 765625.

Some larger examples :