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Duffinian n.  and  pentagonal n.

First numbers that belong to both sets:

35, 247, 425, 651, 715, 925, 1247, 2035, 2501, 3577, 4187, 5735, 6305, 7957, 8177, 8855, 10795, 11837, 12927, 15251, 17767, 18095, 19097, 21901, 22265, 23375, 24897, 26467, 28085, 30175, 31901, 33227, 36895, 38801, 42757, 44807, 47437, 49051, 51245, 53487, 55777, 60501, 61105, 62935, 65417, 73151, 75825, 78547, 79235, 81317, 84847, 87001, 89915, 93625, 95887, 98945, 102835, 105205, 108407, 114955, 118301, 125137, 128627, 133057, 136655, 139385, 143067, 146797, 155365, 158275, 162197, 166167, 170185, 174251, 179401, 182527, 186737, 192067, 195301, 199655, 208507, 209627, 217551, 222145, 223301, 226787, 236215, 241001, 250717, 255647, 261877, 265651, 266915, 281017, 292825, 302177, 313045, 318551, 325501, 348245, 352595, 358437, 364327, 382285, 383801, 388367, 394497, 396037, 406901, 413175, 414751, 419497, 421085, 425867, 438751, 451827, 453475, 458437, 460097, 465095, 473485, 485357, 499105, 520087, 527177, 536107, 541501, 543305, 548735, 563347, 570725, 580015, 585625, 593147, 600717, 616001, 625651, 639287, 655051, 657035, 679057, 681077, 711737, 736751, 747301, 753667, 762197, 779401, 781565, 805567, 807767, 841127, 859195, 868301, 886657, 895907, 914551, 923945, 942877, 964405, 971635, 981317, 1003277, 1010651, 1030447, 1040417, 1063025, 1080777, 1090987, 1101245, 1114135, 1121905, 1132307, 1153255, 1163801, 1185037, 1217251, 1219955, 1249897, 1271901, 1294097, 1316485, 1327751, 1350427, 1361837, 1376167, 1387685, 1396355, 1422527, 1431305, 1443051, 1454845, 1466687, 1478577, 1481557, 1490515, 1493507, 1526617, 1538747, 1553977, 1563151, 1566215, 1587747, 1600117, 1603217, 1612535, 1628125, 1650077, 1662687, 1665847, 1675345, 1678517, 1688051, 1704001, 1713607, 1726457, 1742587, 1752301, 1765295, 1791427, 1794707, 1804565, 1817751, 1830985, 1834301, 1844267, 1860937, 1870975, 1884401, 1911397, 1924967, 1952251, 1955675, 2021301, 2035255, 2038751, 2077405, 2091551, 2119987, 2134277, 2163001, 2206447, 2210087, 2221025, 2239315, 2265047, 2294635, 2298347, 2309501, 2328151, 2339377, 2343125, 2354387, 2373217, 2388335, 2399705, 2418715, 2430157, 2445455, 2476195, 2491637, 2526557, 2538251, 2585297, 2605027, 2616901, 2620865, 2632775, 2664667, 2680685, 2700775, 2716901, 2729027, 2749297, 2777801, 2798251, 2810557, 2827007, 2860051, 2876645.

Some larger examples :