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pentagonal n.  and  pernicious n.

First numbers that belong to both sets:

5, 12, 22, 35, 70, 117, 145, 176, 247, 376, 425, 477, 532, 590, 651, 782, 852, 925, 1001, 1335, 1426, 1617, 1717, 2147, 2380, 2882, 3151, 3290, 3725, 4510, 5017, 6112, 6305, 6501, 7315, 8855, 10292, 10542, 10795, 11310, 11572, 12376, 12650, 13490, 13776, 15862, 16485, 16801, 17120, 18760, 19097, 21540, 24130, 24897, 25285, 27270, 27676, 28912, 29330, 33227, 33675, 34126, 35497, 35960, 36426, 36895, 38320, 41750, 42757, 43265, 43776, 46376, 50692, 51245, 58115, 59900, 63551, 65417, 69876, 73815, 82017, 82720, 84847, 86280, 88452, 91390, 93625, 94376, 95887, 98945, 99717, 104412, 105205, 107602, 109215, 114955, 120842, 121695, 122551, 123410, 131276, 132165, 133057, 133952, 134850, 135751, 139385, 143067, 146797, 155365, 161212, 163185, 164176, 165170, 171197, 172212, 178365, 180440, 187797, 189926, 193142, 194220, 196385, 199655, 202952, 209627, 217551, 229126, 238602, 241001, 243412, 248270, 255647, 266915, 269452, 274562, 278426, 284926, 295482, 296815, 300832, 308947, 315792, 317170, 318551, 329707, 335357, 338200, 341055, 351142, 355510, 361376, 364327, 367290, 373252, 377755, 380772, 382285, 386842, 389895, 394497, 396037, 405340, 406901, 410032, 413175, 424270, 425867, 438751, 440375, 442002, 446901, 450182, 458437, 460097, 463426, 466767, 468442, 473485, 492207, 493927, 495650, 499105, 502572, 504310, 506051, 509542, 514801, 518322, 525400, 527177, 530740, 543305, 545112, 557845, 565187, 567030, 583752, 587501, 593147, 598820, 602617, 617925, 649117, 661012, 673015, 687155, 689187, 693260, 701442, 711737, 717950, 720027, 724190, 726276, 728365, 747301, 749420, 764337, 783732, 790251, 792430, 798985, 803370, 807767, 812176, 818812, 829932, 834401, 847880, 854660, 856926, 859195, 866020, 875162, 879751, 900550, 931022, 933387, 935755, 940500, 942877, 945257, 952415, 964405, 971635, 974051, 986176, 991047, 995930, 998376, 1023001, 1025480, 1032935, 1035426, 1037920, 1050435, 1065552, 1068082, 1080777, 1083325, 1088430, 1096110, 1103817, 1116722, 1129702, 1142757, 1153255, 1155887, 1161160, 1166445, 1179710, 1182372, 1195727, 1222662, 1230801, 1241695, 1244426, 1252637, 1255380.

Some larger examples :