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Harshad n.  and  square n.

First numbers that belong to both sets:

1, 4, 9, 36, 81, 100, 144, 225, 324, 400, 441, 576, 900, 1296, 1521, 1764, 2025, 2304, 2401, 2601, 2704, 2916, 3600, 3969, 4225, 4356, 4624, 5184, 6084, 6400, 7056, 7744, 8100, 9216, 10000, 10404, 10816, 11025, 11664, 12100, 12321, 12544, 13689, 14400, 15876, 17424, 19044, 19600, 20736, 21904, 22500, 24336, 26244, 28224, 30276, 32400, 33124, 36100, 39204, 40000, 40401, 41616, 42849, 44100, 46656, 48400, 54756, 57600, 59049, 60516, 61504, 62001, 63504, 69696, 72900, 75625, 77841, 82944, 88209, 90000, 93636, 99225, 101124, 103041, 104329, 104976, 108900, 110889, 116964, 121104, 122500, 123201, 125316, 129600, 139876, 142884, 152100, 156816, 161604, 164836, 171396, 176400, 186624, 188356, 194481, 198025, 202500, 211600, 213444, 219024, 226576, 227529, 230400, 234256, 236196, 240100, 242064, 244036, 248004, 251001, 254016, 256036, 260100, 263169, 270400, 272484, 278784, 281961, 283024, 286225, 291600, 295936, 297025, 301401, 302500, 304704, 311364, 321489, 324900, 331776, 336400, 352836, 360000, 363609, 374544, 385641, 389376, 390625, 396900, 412164, 419904, 422500, 425104, 431649, 432964, 435600, 442225, 443556, 448900, 455625, 459684, 462400, 465124, 467856, 476100, 480249, 492804, 501264, 504100, 508369, 509796, 518400, 525625, 544644, 558009, 562500, 571536, 577600, 585225, 589824, 592900, 599076, 608400, 613089, 614656, 617796, 627264, 640000, 648025, 656100, 664225, 665856, 670761, 675684, 685584, 695556, 700569, 705600, 715716, 729316, 739600, 746496, 762129, 767376, 774400, 777924, 788544, 799236, 810000, 813604, 819025, 826281, 842724, 853776, 876096, 883600, 887364, 893025, 898704, 910116, 921600, 944784, 956484, 962361, 968256, 970225, 972196, 980100, 984064, 990025, 998001, 1000000, 1004004, 1008016, 1010025, 1014049, 1016064, 1020100, 1022121, 1024144, 1034289, 1040400, 1052676, 1065024, 1073296, 1081600, 1089936, 1092025, 1102500, 1108809, 1115136, 1127844, 1140624, 1155625, 1162084, 1166400, 1177225, 1179396, 1185921, 1205604, 1210000, 1212201, 1221025, 1225449, 1232100, 1245456, 1254400, 1258884, 1265625, 1276900, 1285956, 1306449, 1313316, 1320201, 1327104, 1347921, 1354896, 1357225, 1368900, 1380625.

Some larger examples :