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cyclic n.  and  happy n.

First numbers that belong to both sets:

1, 7, 13, 19, 23, 31, 79, 91, 97, 103, 109, 133, 139, 167, 193, 239, 263, 293, 313, 319, 329, 331, 365, 367, 379, 383, 391, 397, 409, 469, 487, 563, 565, 617, 623, 635, 649, 653, 665, 671, 673, 683, 709, 739, 761, 763, 793, 863, 881, 899, 901, 907, 913, 923, 937, 973, 989, 1003, 1009, 1033, 1039, 1067, 1093, 1115, 1121, 1141, 1151, 1177, 1211, 1247, 1257, 1277, 1285, 1303, 1309, 1333, 1335, 1337, 1339, 1353, 1373, 1393, 1411, 1427, 1447, 1457, 1481, 1487, 1511, 1527, 1547, 1607, 1663, 1679, 1697, 1717, 1727, 1733, 1745, 1769, 1841, 1847, 1851, 1903, 1933, 1959, 2003, 2039, 2063, 2091, 2093, 2111, 2147, 2157, 2177, 2185, 2221, 2245, 2257, 2285, 2309, 2333, 2339, 2369, 2383, 2393, 2417, 2471, 2517, 2545, 2555, 2557, 2571, 2581, 2603, 2685, 2693, 2717, 2741, 2755, 2771, 2815, 2833, 2845, 2851, 2881, 2899, 2903, 2933, 2963, 3001, 3013, 3019, 3029, 3031, 3065, 3067, 3079, 3083, 3091, 3097, 3103, 3109, 3133, 3137, 3139, 3173, 3193, 3209, 3233, 3239, 3269, 3293, 3301, 3313, 3317, 3319, 3323, 3329, 3331, 3365, 3367, 3371, 3383, 3391, 3463, 3531, 3553, 3563, 3605, 3607, 3635, 3637, 3643, 3653, 3661, 3667, 3673, 3689, 3709, 3713, 3731, 3763, 3779, 3797, 3803, 3823, 3833, 3869, 3901, 3907, 3923, 3931, 3977, 4009, 4087, 4111, 4127, 4147, 4157, 4181, 4187, 4217, 4255, 4271, 4285, 4363, 4441, 4447, 4471, 4481, 4517, 4571, 4577, 4633, 4663, 4721, 4741, 4747, 4751, 4757, 4781, 4811, 4817, 4841, 4871, 5063, 5065, 5111, 5127, 5147, 5199, 5227, 5245, 5257, 5281, 5353, 5363, 5417, 5471, 5477, 5485, 5527, 5533, 5545, 5569, 5585, 5603, 5605, 5633, 5659, 5695, 5721, 5741, 5747, 5781, 5821, 5845, 5879, 5897.

Some larger examples :