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Duffinian n.  and  heptagonal n.

First numbers that belong to both sets:

55, 81, 189, 235, 403, 469, 697, 1177, 1525, 1651, 2059, 2839, 3553, 4141, 5221, 6943, 9211, 9517, 12145, 13213, 13579, 14707, 15093, 16281, 16687, 17935, 18361, 19669, 21949, 23377, 23863, 25351, 27931, 29539, 30085, 31753, 34047, 34633, 37027, 38875, 42055, 44023, 44689, 46717, 50197, 53071, 55279, 61387, 62173, 64561, 68641, 71149, 74563, 78943, 81631, 85285, 86211, 89019, 92833, 93799, 96727, 101707, 108889, 114169, 117397, 118483, 121771, 127351, 130759, 131905, 141253, 144841, 146047, 159643, 164737, 171217, 175165, 176491, 180499, 187279, 191407, 192793, 196981, 204061, 208369, 214183, 221563, 226051, 227557, 232105, 244453, 246019, 250747, 258727, 263575, 265201, 270109, 276723, 278389, 283417, 285103, 290191, 298771, 303979, 305725, 310993, 319873, 325261, 332515, 341695, 349129, 354757, 364237, 369985, 371911, 377719, 387499, 393427, 395413, 403407, 411481, 417589, 425803, 436183, 442471, 444577, 450925, 468073, 476767, 478953, 487747, 496621, 503329, 514609, 521437, 523723, 530611, 549199, 551545, 570493, 577681, 580087, 599515, 606883, 609349, 616777, 629257, 636805, 639331, 646939, 657153, 659719, 667447, 677821, 690901, 698809, 701455, 709423, 722803, 730891, 733597, 752679, 755425, 763693, 766459, 774787, 788767, 797215, 800041, 808549, 834343, 857611, 866419, 878233, 902101, 914155, 929335, 938503, 941569, 950797, 953883, 978751, 988159, 1003939, 1016653, 1026241, 1052029, 1055275, 1065043, 1078137, 1081423, 1091311, 1094617, 1121245, 1131313, 1144807, 1161787, 1175461, 1185769, 1203049, 1213477, 1216963, 1227451, 1245031, 1255639, 1259185, 1287733, 1298521, 1302127, 1316601, 1327509, 1331155, 1342123, 1345789, 1375297, 1390171, 1401379, 1420159, 1435273, 1446661, 1465741, 1477249, 1481095, 1492663, 1523731, 1527637, 1539385, 1574899, 1586827, 1606807, 1618855, 1622881, 1634989, 1667497, 1671583, 1704451, 1716859, 1733473, 1737639, 1754353, 1766941, 1783795, 1817743, 1834837, 1856317, 1869265, 1873591, 1886599, 1904013, 1908379, 1921507, 1925893, 1939081, 1961161, 1974469, 1992283, 2014663, 2028151, 2032657, 2082553, 2087119, 2100847, 2137675, 2142301, 2156209, 2193517, 2198203, 2212291, 2216997, 2250079, 2269093, 2292973, 2307361, 2326615, 2350795, 2370229, 2384857, 2389743, 2404431, 2409337.

Some larger examples :