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cyclic n.  and  nonagonal n.

First numbers that belong to both sets:

1, 559, 1639, 3729, 3961, 4699, 5781, 7291, 11629, 12871, 16491, 23329, 30691, 32689, 41311, 44409, 48439, 69231, 70219, 77331, 85879, 96031, 109209, 110449, 114211, 129889, 142309, 158259, 159751, 178201, 182971, 197659, 211191, 239599, 262081, 267859, 277629, 285571, 310069, 342109, 362089, 389611, 396649, 418141, 425431, 447679, 486019, 509779, 539589, 542341, 550639, 561801, 584461, 619291, 655129, 682669, 720271, 729829, 758881, 798499, 849439, 905511, 923401, 934219, 967051, 1007949, 1011709, 1023031, 1084479, 1104049, 1147719, 1151731, 1163809, 1200421, 1212751, 1250119, 1262701, 1313659, 1330869, 1352539, 1405261, 1423059, 1458991, 1472581, 1490799, 1527571, 1583569, 1607199, 1621461, 1626229, 1683991, 1698589, 1742761, 1757611, 1802539, 1878679, 1925119, 1982649, 1987921, 2003779, 2025021, 2051731, 2067841, 2089419, 2116549, 2132911, 2138379, 2160321, 2198989, 2221239, 2249209, 2288661, 2317051, 2334169, 2385901, 2403271, 2455759, 2544499, 2598499, 2671381, 2689759, 2745271, 2820169, 2915209, 2972989, 3050911, 3070549, 3096831, 3129841, 3149731, 3176349, 3203079, 3209779, 3372679, 3476539, 3539611, 3560761, 3589059, 3624589, 3645991, 3732229, 3739461, 3768459, 3797569, 3848781, 3885571, 3974581, 3996991, 4057059, 4178539, 4247659, 4340701, 4364119, 4403289, 4434751, 4458421, 4553731, 4617831, 4650049, 4722949, 4821031, 4920121, 4945051, 5045401, 5146759, 5223439, 5283471, 5326561, 5352499, 5456881, 5527029, 5562271, 5668669, 5749129, 5848239, 5857291, 5884489, 5966461, 5993911, 6076639, 6104341, 6178521, 6215779, 6300019, 6403749, 6413221, 6527431, 6594519, 6671611, 6797841, 6876109, 6994351, 7024069, 7143571, 7233859, 7385599, 7477399, 7590369, 7724971, 7756201, 7850269, 7881751, 7934361, 7966011, 8008309, 8103889, 8232211, 8264449, 8361541, 8394031, 8437449, 8888941, 8922439, 9057061, 9113451, 9158689, 9329329, 9375099, 9432469, 9570871, 9605629, 9745291, 9850699, 9944871.