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cyclic n.  and  Friedman n.

First numbers that belong to both sets:

127, 347, 1285, 2501, 2503, 2505, 2507, 2509, 2737, 3281, 6145, 10255, 11663, 11665, 12091, 12101, 12107, 12109, 13243, 13813, 13823, 14129, 14647, 15567, 15613, 15617, 15629, 15635, 15641, 15661, 15667, 15679, 15689, 15697, 16295, 16381, 16385, 16387, 16447, 16759, 16765, 16783, 16797, 16879, 17437, 18435, 19653, 19693, 19739, 21495, 21845, 21943, 21953, 22757, 24385, 24393, 25895, 27653, 28547, 28559, 29517, 29519, 29527, 29531, 29549, 32747, 32759, 32765, 32771, 32783, 32785, 32795, 32815, 32845, 32849, 32853, 32885, 33655, 34991, 35933, 36457, 37187, 38427, 39283, 39313, 39343, 39363, 39369, 39945, 43691, 44977, 45361, 45367, 45947, 46613, 46615, 46619, 46633, 46637, 46643, 46649, 46651, 46653, 46657, 46661, 46663, 46673, 46691, 48751, 48757, 48759, 49277, 52429, 52483, 52493, 53245, 53297, 54953, 58921, 58957, 58971, 59039, 59051, 59053, 59057, 59059, 59129, 59147, 59263, 59273, 59313, 59759, 62503, 63895, 64513, 65471, 65483, 65487, 65489, 65491, 65531, 69253, 69895, 69985, 74183, 74353, 74533, 74897, 75433, 77459, 78055, 78133, 78163, 78545, 78605, 78659, 83357, 83957, 91853, 97333, 97343, 98435, 101299, 102391, 102497, 102505, 102955, 103681, 103685, 103687, 104879, 104897, 104969, 105295, 106289, 106923, 107329, 108233, 109351, 109357, 112895, 114245, 114377, 114395, 114647, 114679, 114697, 114737, 114779, 115249, 115699, 116317, 116635, 116729, 116827, 116837, 117067, 117571, 117577, 117587, 117613, 117619, 117627, 117629, 117633, 117641, 117643, 117647, 117653, 117659, 117661, 117667, 117673, 117677, 117685, 117695, 117697, 117739, 117763, 117767, 117785, 118259, 118327, 118543, 118567, 118673, 122503, 122791, 123199, 123265, 124385, 124415, 124483, 124585, 124973, 124991, 125003, 125005, 125007, 125009, 125011, 125017, 125021, 125023, 125029, 125031, 125033, 125039, 125041, 125043, 125047, 125051, 125053, 125057, 125059, 125061, 125063, 125065, 125067, 125071, 125077, 125081, 125083, 125087, 125089, 125093, 125095, 125099, 125317, 125943, 125995, 126035, 126397, 126559, 126563, 126737, 127417, 127453, 127583, 127587.

Some larger examples :