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evil n.  and  Proth n.

First numbers that belong to both sets:

3, 5, 9, 17, 33, 57, 65, 113, 129, 177, 209, 225, 257, 353, 417, 449, 513, 609, 673, 705, 801, 833, 897, 993, 1025, 1217, 1345, 1409, 1601, 1665, 1793, 1985, 2049, 2241, 2369, 2433, 2625, 2689, 2817, 3009, 3137, 3201, 3329, 3521, 3585, 3777, 3905, 3969, 4097, 4481, 4737, 4865, 5249, 5377, 5633, 6017, 6273, 6401, 6657, 7041, 7169, 7553, 7809, 7937, 8193, 8577, 8833, 8961, 9345, 9473, 9729, 10113, 10369, 10497, 10753, 11137, 11265, 11649, 11905, 12033, 12417, 12545, 12801, 13185, 13313, 13697, 13953, 14081, 14337, 14721, 14977, 15105, 15489, 15617, 15873, 16257, 16385, 17153, 17665, 17921, 18689, 18945, 19457, 20225, 20737, 20993, 21505, 22273, 22529, 23297, 23809, 24065, 24833, 25089, 25601, 26369, 26625, 27393, 27905, 28161, 28673, 29441, 29953, 30209, 30977, 31233, 31745, 32513, 32769, 33537, 34049, 34305, 35073, 35329, 35841, 36609, 37121, 37377, 37889, 38657, 38913, 39681, 40193, 40449, 41217, 41473, 41985, 42753, 43009, 43777, 44289, 44545, 45057, 45825, 46337, 46593, 47361, 47617, 48129, 48897, 49409, 49665, 50177, 50945, 51201, 51969, 52481, 52737, 53249, 54017, 54529, 54785, 55553, 55809, 56321, 57089, 57345, 58113, 58625, 58881, 59649, 59905, 60417, 61185, 61697, 61953, 62465, 63233, 63489, 64257, 64769, 65025, 65537, 67073, 68097, 68609, 70145, 70657, 71681, 73217, 74241, 74753, 75777, 77313, 77825, 79361, 80385, 80897, 82433, 82945, 83969, 85505, 86017, 87553, 88577, 89089, 90113, 91649, 92673, 93185, 94721, 95233, 96257, 97793, 98817, 99329, 100353, 101889, 102401, 103937, 104961, 105473, 106497, 108033, 109057, 109569, 111105, 111617, 112641, 114177, 114689, 116225, 117249, 117761, 119297, 119809, 120833, 122369, 123393, 123905, 124929, 126465, 126977, 128513, 129537, 130049, 131073, 132609, 133633, 134145, 135681, 136193, 137217, 138753, 139777, 140289, 141313, 142849, 143361, 144897, 145921, 146433, 147969, 148481, 149505, 151041, 151553, 153089, 154113, 154625.

Some larger examples :