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amenable n.  and  pentagonal n.

First numbers that belong to both sets:

1, 5, 12, 92, 117, 145, 176, 376, 425, 477, 532, 852, 925, 1001, 1080, 1520, 1617, 1717, 1820, 2380, 2501, 2625, 2752, 3432, 3577, 3725, 3876, 4676, 4845, 5017, 5192, 6112, 6305, 6501, 6700, 7740, 7957, 8177, 8400, 9560, 9801, 10045, 10292, 11572, 11837, 12105, 12376, 13776, 14065, 14357, 14652, 16172, 16485, 16801, 17120, 18760, 19097, 19437, 19780, 21540, 21901, 22265, 22632, 24512, 24897, 25285, 25676, 27676, 28085, 28497, 28912, 31032, 31465, 31901, 32340, 34580, 35037, 35497, 35960, 38320, 38801, 39285, 39772, 42252, 42757, 43265, 43776, 46376, 46905, 47437, 47972, 50692, 51245, 51801, 52360, 55200, 55777, 56357, 56940, 59900, 60501, 61105, 61712, 64792, 65417, 66045, 66676, 69876, 70525, 71177, 71832, 75152, 75825, 76501, 77180, 80620, 81317, 82017, 82720, 86280, 87001, 87725, 88452, 92132, 92877, 93625, 94376, 98176, 98945, 99717, 100492, 104412, 105205, 106001, 106800, 110840, 111657, 112477, 113300, 117460, 118301, 119145, 119992, 124272, 125137, 126005, 126876, 131276, 132165, 133057, 133952, 138472, 139385, 140301, 141220, 145860, 146797, 147737, 148680, 153440, 154401, 155365, 156332, 161212, 162197, 163185, 164176, 169176, 170185, 171197, 172212, 177332, 178365, 179401, 180440, 185680, 186737, 187797, 188860, 194220, 195301, 196385, 197472, 202952, 204057, 205165, 206276, 211876, 213005, 214137, 215272, 220992, 222145, 223301, 224460, 230300, 231477, 232657, 233840, 239800, 241001, 242205, 243412, 249492, 250717, 251945, 253176, 259376, 260625, 261877, 263132, 269452, 270725, 272001, 273280, 279720, 281017, 282317, 283620, 290180, 291501, 292825, 294152, 300832, 302177, 303525, 304876, 311676, 313045, 314417, 315792, 322712, 324105, 325501, 326900, 333940, 335357, 336777, 338200, 345360, 346801, 348245, 349692, 356972, 358437, 359905, 361376, 368776, 370265, 371757, 373252, 380772, 382285, 383801, 385320, 392960, 394497, 396037, 397580, 405340, 406901, 408465, 410032, 417912, 419497, 421085, 422676, 430676, 432285, 433897, 435512, 443632, 445265, 446901, 448540, 456780, 458437, 460097, 461760, 470120.

Some larger examples :