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Duffinian n.  and  octagonal n.

First numbers that belong to both sets:

8, 21, 65, 133, 225, 341, 481, 833, 1281, 1541, 1825, 2133, 3605, 4033, 4961, 5461, 6533, 8321, 8965, 11041, 12545, 13333, 14981, 15841, 16725, 17633, 19521, 20501, 21505, 25761, 28033, 30401, 31621, 34133, 36741, 38081, 42245, 46625, 48133, 51221, 52801, 56033, 68101, 71765, 73633, 79381, 83333, 85345, 87381, 89441, 91525, 97921, 100101, 109061, 111361, 116033, 120801, 123221, 125665, 133141, 138245, 140833, 146081, 148741, 151425, 162401, 168033, 173761, 176661, 179585, 182533, 188501, 191521, 197633, 206981, 210145, 219781, 226325, 229633, 236321, 249985, 253461, 256961, 271201, 278465, 282133, 285825, 293281, 300833, 312341, 324065, 328021, 332001, 340033, 348161, 360533, 368901, 373121, 377365, 390241, 394581, 398945, 403333, 412181, 416641, 425633, 434721, 439301, 443905, 448533, 457861, 472033, 481601, 486421, 496133, 510881, 515845, 520833, 530881, 546133, 556421, 561601, 572033, 587861, 625633, 636641, 642181, 653333, 658945, 704705, 710533, 722261, 728161, 740033, 758021, 782341, 794645, 800833, 819541, 832133, 838465, 851201, 876961, 883461, 896533, 909701, 916321, 929633, 936325, 949781, 956545, 963333, 976981, 990725, 1018501, 1032533, 1053761, 1060885, 1068033, 1082401, 1104133, 1118741, 1126081, 1140833, 1163141, 1170625, 1200801, 1208405, 1231361, 1239061, 1254533, 1270101, 1277921, 1285765, 1309441, 1333333, 1349381, 1357441, 1373633, 1398101, 1406305, 1431061, 1456033, 1472801, 1515141, 1523681, 1532245, 1558081, 1584133, 1601621, 1610401, 1628033, 1645761, 1654661, 1690501, 1708565, 1717633, 1735841, 1744981, 1791041, 1828321, 1856533, 1875461, 1884961, 1904033, 1932821, 1971541, 1981281, 2000833, 2020481, 2030341, 2050133, 2080001, 2090005, 2100033, 2120161, 2170901, 2181121, 2201633, 2232581, 2242945, 2284641, 2295125, 2305633, 2326721, 2337301, 2358533, 2390561, 2444421, 2455265, 2466133, 2487941, 2520833, 2553941, 2576133, 2620805, 2632033, 2665861, 2688533, 2722721, 2768641, 2803333, 2826581, 2861633, 2896901, 2908705, 2932385, 2944261, 2956161, 2980033, 3016021, 3040133, 3076481, 3125281, 3162133, 3186821, 3199201, 3224033, 3261461, 3286533, 3336965, 3349633, 3375041, 3387781, 3451841, 3490565, 3503521, 3529505, 3542533, 3568661, 3581761, 3608033, 3634401, 3647621, 3660865, 3700741, 3727445, 3740833, 3767681, 3808133.

Some larger examples :