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cyclic n.  and  octagonal n.

First numbers that belong to both sets:

1, 65, 133, 341, 481, 1541, 2465, 2821, 3605, 4033, 5461, 6533, 8321, 11041, 12545, 13333, 14981, 15841, 17633, 18565, 20501, 23585, 24661, 26885, 31621, 34133, 38081, 48133, 51221, 52801, 56033, 57685, 61061, 62785, 66305, 68101, 73633, 77441, 79381, 83333, 89441, 95765, 97921, 102305, 104533, 109061, 111361, 116033, 118405, 133141, 138245, 140833, 146081, 148741, 154133, 156865, 162401, 165205, 173761, 182533, 185505, 188501, 197633, 206981, 213333, 216545, 219781, 229633, 246533, 256961, 260485, 267605, 282133, 293281, 300833, 312341, 328021, 336005, 340033, 348161, 360533, 373121, 398945, 403333, 412181, 416641, 425633, 430165, 439301, 453185, 457861, 462561, 472033, 481601, 486421, 510881, 530881, 546133, 556421, 561601, 577285, 587861, 603905, 609301, 636641, 642181, 653333, 670241, 704705, 710533, 722261, 746005, 758021, 782341, 794645, 800833, 813281, 819541, 832133, 851201, 857605, 876961, 896533, 909701, 916321, 929633, 949781, 956545, 983841, 997633, 1011521, 1018501, 1032533, 1053761, 1060885, 1075205, 1082401, 1104133, 1118741, 1126081, 1148245, 1163141, 1185665, 1193221, 1231361, 1239061, 1254533, 1277921, 1309441, 1325345, 1333333, 1349381, 1357441, 1373633, 1398101, 1406305, 1431061, 1447685, 1456033, 1464405, 1472801, 1498133, 1523681, 1558081, 1584133, 1636885, 1654661, 1690501, 1717633, 1735841, 1744981, 1791041, 1828321, 1884961, 1904033, 1913605, 1932821, 1952133, 1971541, 1991045, 2000833, 2020481, 2030341, 2050133, 2090005, 2110085, 2170901, 2181121, 2201633, 2232581, 2242945, 2274181, 2305633, 2326721, 2337301, 2358533, 2390561, 2401285, 2433601, 2466133, 2487941, 2520833, 2531845, 2553941, 2587265, 2654561, 2688533, 2699905, 2722721, 2803333, 2826581, 2838241, 2861633, 2908705, 2932385, 2944261, 2968085, 3004001, 3016021, 3040133, 3076481, 3113045, 3125281, 3186821, 3199201, 3224033, 3236485, 3261461, 3299105, 3311701, 3349633, 3375041, 3387781, 3413333, 3451841, 3490565, 3542533, 3568661, 3581761, 3647621, 3660865, 3700741, 3727445, 3740833, 3767681, 3781141, 3821665, 3848801, 3889685, 3903361, 3930785, 3944533, 3985921, 4013633, 4055381, 4069345, 4111381, 4153633, 4181921, 4224533, 4281685, 4324801, 4397141, 4411681, 4440833, 4455445, 4484741, 4543621, 4588033, 4632661, 4662533, 4677505, 4767841, 4843781, 4904965, 4935701, 4951105, 4981985, 5044033, 5075201.

Some larger examples :