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Duffinian n.  and  Proth n.

First numbers that belong to both sets:

9, 25, 49, 57, 65, 81, 129, 161, 209, 225, 289, 385, 417, 481, 513, 737, 833, 865, 961, 993, 1025, 1089, 1281, 1537, 1793, 1857, 1921, 1985, 2177, 2305, 2369, 2433, 2497, 2561, 2817, 2881, 3073, 3265, 3649, 3713, 3841, 3905, 4033, 4097, 4225, 4609, 4737, 5249, 5377, 5633, 5761, 5889, 6017, 6401, 6913, 7041, 7169, 8065, 8193, 8321, 8449, 8705, 8833, 9089, 9217, 9985, 10497, 10625, 10881, 11009, 11137, 11521, 11905, 12545, 12673, 13953, 14209, 14849, 14977, 15745, 16129, 16257, 16897, 17153, 17409, 17665, 18689, 19713, 19969, 20737, 20993, 21249, 21505, 21761, 22529, 23553, 23809, 24065, 24321, 24577, 24833, 25345, 26369, 27137, 27649, 27905, 28417, 28673, 28929, 29441, 29953, 30209, 31745, 32513, 33025, 33281, 33537, 33793, 34049, 34561, 34817, 35073, 35329, 35585, 35841, 36865, 37121, 37377, 38401, 38657, 39169, 40449, 40705, 41473, 42241, 42753, 43009, 44801, 45313, 46081, 46849, 47105, 47361, 47617, 47873, 48385, 48641, 49153, 50689, 50945, 51201, 51457, 52481, 52993, 53249, 53761, 54017, 54273, 54529, 54785, 55297, 55553, 56065, 56321, 56577, 56833, 57857, 58625, 58881, 59905, 60673, 60929, 61697, 62209, 62465, 62977, 63233, 63489, 64769, 65025, 65281, 66049, 68097, 68609, 69121, 71681, 72193, 73217, 73729, 75265, 77313, 77825, 78337, 78849, 79361, 82945, 86529, 88577, 90113, 91649, 92161, 92673, 93185, 93697, 96257, 97281, 97793, 98305, 99329, 100865, 103937, 104449, 105473, 105985, 106497, 107521, 108545, 109057, 109569, 110081, 110593, 111617, 114177, 116737, 117761, 121345, 122369, 122881, 123393, 123905, 124417, 124929, 125953, 127489, 128001, 128513, 129025, 130049, 130561, 131585, 132609, 134657, 135169, 137729, 138753, 139265, 140801, 141313, 142337, 142849, 144385, 145409, 145921, 146433, 148481, 148993, 150017, 150529, 153601, 154625, 156161, 156673, 157697, 159745, 160257, 160769, 161793, 162305, 162817, 164353, 165377, 165889, 166401, 166913, 167425, 168961, 169985, 171521, 172033, 173057.

Some larger examples :