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amenable n.  and  nonagonal n.

First numbers that belong to both sets:

1, 9, 24, 204, 261, 325, 396, 856, 969, 1089, 1216, 1956, 2125, 2301, 2484, 3504, 3729, 3961, 4200, 5500, 5781, 6069, 6364, 7944, 8281, 8625, 8976, 10836, 11229, 11629, 12036, 14176, 14625, 15081, 15544, 17964, 18469, 18981, 19500, 22200, 22761, 23329, 23904, 26884, 27501, 28125, 28756, 32016, 32689, 33369, 34056, 37596, 38325, 39061, 39804, 43624, 44409, 45201, 46000, 50100, 50941, 51789, 52644, 57024, 57921, 58825, 59736, 64396, 65349, 66309, 67276, 72216, 73225, 74241, 75264, 80484, 81549, 82621, 83700, 89200, 90321, 91449, 92584, 98364, 99541, 100725, 101916, 107976, 109209, 110449, 111696, 118036, 119325, 120621, 121924, 128544, 129889, 131241, 132600, 139500, 140901, 142309, 143724, 150904, 152361, 153825, 155296, 162756, 164269, 165789, 167316, 175056, 176625, 178201, 179784, 187804, 189429, 191061, 192700, 201000, 202681, 204369, 206064, 214644, 216381, 218125, 219876, 228736, 230529, 232329, 234136, 243276, 245125, 246981, 248844, 258264, 260169, 262081, 264000, 273700, 275661, 277629, 279604, 289584, 291601, 293625, 295656, 305916, 307989, 310069, 312156, 322696, 324825, 326961, 329104, 339924, 342109, 344301, 346500, 357600, 359841, 362089, 364344, 375724, 378021, 380325, 382636, 394296, 396649, 399009, 401376, 413316, 415725, 418141, 420564, 432784, 435249, 437721, 440200, 452700, 455221, 457749, 460284, 473064, 475641, 478225, 480816, 493876, 496509, 499149, 501796, 515136, 517825, 520521, 523224, 536844, 539589, 542341, 545100, 559000, 561801, 564609, 567424, 581604, 584461, 587325, 590196, 604656, 607569, 610489, 613416, 628156, 631125, 634101, 637084, 652104, 655129, 658161, 661200, 676500, 679581, 682669, 685764, 701344, 704481, 707625, 710776, 726636, 729829, 733029, 736236, 752376, 755625, 758881, 762144, 778564, 781869, 785181, 788500, 805200, 808561, 811929, 815304, 832284, 835701, 839125, 842556, 859816, 863289, 866769, 870256, 887796, 891325, 894861, 898404, 916224, 919809, 923401, 927000, 945100, 948741, 952389, 956044, 974424, 978121, 981825, 985536, 1004196, 1007949, 1011709, 1015476, 1034416, 1038225, 1042041, 1045864, 1065084, 1068949, 1072821, 1076700, 1096200.

Some larger examples :