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centered octagonal n.  and  frugal n.

First numbers that belong to both sets:

625, 729, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18225, 18769, 19321, 21609, 22201, 22801, 24649, 26569, 27889, 28561, 29929, 30625, 32041, 32761, 35721, 36481, 37249, 38809, 39601, 44521, 49729, 50625, 51529, 52441, 54289, 57121, 58081, 59049, 60025, 63001, 66049, 69169, 72361, 73441, 76729, 78961, 80089, 83521, 85849, 94249, 96721, 97969, 100489, 105625, 108241, 109561, 110889, 112225, 113569, 117649, 120409, 121801, 123201, 124609, 126025, 128881, 130321, 131769, 133225, 134689, 136161, 137641, 139129, 140625, 143641, 146689, 149769, 151321, 156025, 157609, 160801, 164025, 167281, 170569, 172225, 175561, 177241, 178929, 180625, 182329, 185761, 187489, 192721, 194481, 196249, 198025, 201601, 208849, 210681, 212521, 214369, 218089, 219961, 225625, 227529, 229441, 235225, 237169, 241081, 247009, 249001, 253009, 257049, 259081, 261121, 263169, 271441, 273529, 279841, 281961, 290521, 292681, 299209, 301401, 305809, 310249, 316969, 321489, 323761, 326041, 330625, 332929, 337561, 344569, 351649, 358801, 361201, 363609, 366025, 368449, 375769, 380689, 383161, 385641, 388129, 390625, 398161, 405769, 408321, 410881, 413449, 418609, 426409, 431649, 434281, 436921, 452929, 455625, 458329, 461041, 466489, 477481, 491401, 502681, 505521, 516961, 525625, 528529, 531441, 537289, 546121, 552049, 558009, 564001, 573049, 579121, 591361, 597529, 600625, 613089, 619369, 635209, 641601, 654481, 657721, 674041, 677329, 683929, 687241, 693889, 700569, 703921, 707281, 714025, 717409, 727609, 734449, 737881, 744769, 751689, 762129, 765625, 769129, 776161, 779689, 786769, 793881, 822649, 829921, 844561, 855625, 863041, 866761, 877969, 885481, 896809, 908209, 923521, 935089, 942841, 954529, 966289, 982081, 994009, 998001, 1006009, 1014049, 1018081, 1022121, 1026169, 1034289, 1038361, 1042441, 1050625, 1054729, 1058841, 1062961, 1067089, 1075369, 1079521, 1083681, 1087849, 1096209, 1100401, 1104601, 1108809, 1113025, 1117249, 1121481, 1125721, 1129969, 1138489, 1142761, 1151329, 1155625, 1159929, 1164241, 1168561, 1172889, 1181569, 1185921, 1190281, 1194649.

Some larger examples :