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Friedman n.  and  square n.

First numbers that belong to both sets:

25, 121, 289, 625, 1024, 1296, 2500, 2916, 4096, 4624, 5776, 9025, 9216, 10201, 11025, 11664, 12100, 12321, 12544, 12769, 12996, 13225, 13689, 14641, 15625, 16384, 17689, 18225, 19321, 20736, 24336, 24964, 25921, 26244, 26896, 27889, 28224, 28561, 28900, 29584, 29929, 30625, 32761, 35721, 36864, 37249, 38416, 42025, 43264, 45369, 46656, 53824, 55225, 59049, 65536, 68644, 70225, 73984, 81225, 82944, 83521, 85264, 99225, 103684, 104329, 104976, 105625, 106929, 107584, 108241, 112896, 114244, 114921, 117649, 118336, 119025, 123904, 124609, 125316, 126025, 126736, 127449, 128164, 129600, 130321, 131769, 132496, 133225, 133956, 134689, 135424, 136161, 137641, 139129, 140625, 141376, 142129, 142884, 145924, 146689, 147456, 148225, 153664, 159201, 162409, 163216, 164025, 167281, 169744, 170569, 171396, 176400, 177241, 178929, 180625, 181476, 183184, 186624, 188356, 189225, 192721, 194481, 195364, 196249, 198025, 207936, 209764, 210681, 214369, 215296, 216225, 217156, 219024, 219961, 221841, 222784, 223729, 224676, 225625, 226576, 227529, 228484, 231361, 232324, 233289, 234256, 235225, 236196, 237169, 238144, 239121, 241081, 242064, 243049, 244036, 247009, 250000, 251001, 252004, 253009, 254016, 255025, 256036, 257049, 258064, 259081, 262144, 263169, 264196, 265225, 267289, 268324, 269361, 271441, 272484, 273529, 274576, 275625, 278784, 279841, 281961, 283024, 285156, 286225, 287296, 288369, 290521, 291600, 292681, 293764, 294849, 295936, 297025, 298116, 314721, 319225, 320356, 321489, 322624, 327184, 328329, 329476, 331776, 334084, 338724, 342225, 345744, 346921, 351649, 352836, 354025, 355216, 361201, 362404, 363609, 366025, 367236, 376996, 378225, 379456, 381924, 385641, 388129, 389376, 390625, 391876, 393129, 398161, 401956, 412164, 414736, 416025, 417316, 419904, 426409, 427716, 432964, 436921, 442225, 446224, 454276, 455625, 456976, 458329, 462400, 465124, 467856, 469225, 470596, 471969, 474721, 478864, 487204, 495616, 497025, 511225, 512656, 521284, 522729, 524176, 525625, 527076, 531441, 532900, 535824, 537289, 570025, 571536, 577600, 579121, 582169, 585225, 586756, 589824, 597529, 602176, 614656, 616225, 627264, 632025.

Some larger examples :