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amenable n.  and  hexagonal n.

First numbers that belong to both sets:

1, 28, 45, 120, 153, 276, 325, 496, 561, 780, 861, 1128, 1225, 1540, 1653, 2016, 2145, 2556, 2701, 3160, 3321, 3828, 4005, 4560, 4753, 5356, 5565, 6216, 6441, 7140, 7381, 8128, 8385, 9180, 9453, 10296, 10585, 11476, 11781, 12720, 13041, 14028, 14365, 15400, 15753, 16836, 17205, 18336, 18721, 19900, 20301, 21528, 21945, 23220, 23653, 24976, 25425, 26796, 27261, 28680, 29161, 30628, 31125, 32640, 33153, 34716, 35245, 36856, 37401, 39060, 39621, 41328, 41905, 43660, 44253, 46056, 46665, 48516, 49141, 51040, 51681, 53628, 54285, 56280, 56953, 58996, 59685, 61776, 62481, 64620, 65341, 67528, 68265, 70500, 71253, 73536, 74305, 76636, 77421, 79800, 80601, 83028, 83845, 86320, 87153, 89676, 90525, 93096, 93961, 96580, 97461, 100128, 101025, 103740, 104653, 107416, 108345, 111156, 112101, 114960, 115921, 118828, 119805, 122760, 123753, 126756, 127765, 130816, 131841, 134940, 135981, 139128, 140185, 143380, 144453, 147696, 148785, 152076, 153181, 156520, 157641, 161028, 162165, 165600, 166753, 170236, 171405, 174936, 176121, 179700, 180901, 184528, 185745, 189420, 190653, 194376, 195625, 199396, 200661, 204480, 205761, 209628, 210925, 214840, 216153, 220116, 221445, 225456, 226801, 230860, 232221, 236328, 237705, 241860, 243253, 247456, 248865, 253116, 254541, 258840, 260281, 264628, 266085, 270480, 271953, 276396, 277885, 282376, 283881, 288420, 289941, 294528, 296065, 300700, 302253, 306936, 308505, 313236, 314821, 319600, 321201, 326028, 327645, 332520, 334153, 339076, 340725, 345696, 347361, 352380, 354061, 359128, 360825, 365940, 367653, 372816, 374545, 379756, 381501, 386760, 388521, 393828, 395605, 400960, 402753, 408156, 409965, 415416, 417241, 422740, 424581, 430128, 431985, 437580, 439453, 445096, 446985, 452676, 454581, 460320, 462241, 468028, 469965, 475800, 477753, 483636, 485605, 491536, 493521, 499500, 501501, 507528, 509545, 515620, 517653, 523776, 525825, 531996, 534061, 540280, 542361, 548628, 550725, 557040, 559153, 565516, 567645, 574056, 576201, 582660, 584821, 591328, 593505, 600060, 602253, 608856, 611065, 617716, 619941, 626640.

Some larger examples :