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brilliant n.  and  Proth n.

First numbers that belong to both sets:

9, 25, 49, 209, 289, 481, 737, 961, 1537, 2881, 3649, 3713, 11009, 16129, 28417, 45313, 48641, 50689, 52993, 61697, 64769, 66049, 73217, 88577, 104449, 130561, 141313, 145921, 181249, 197633, 199681, 201217, 215041, 219137, 222721, 227329, 234497, 237569, 259073, 264193, 273409, 353281, 356353, 391169, 412673, 434177, 486401, 529409, 532481, 537601, 654337, 1308673, 1861633, 2168833, 2191361, 2193409, 2426881, 2439169, 2523137, 2703361, 2879489, 2899969, 3143681, 3315713, 3340289, 3409921, 3440641, 3495937, 3575809, 3788801, 3790849, 4022273, 4059137, 4349953, 4374529, 4489217, 4546561, 4571137, 4833281, 4964353, 5025793, 5103617, 5148673, 5275649, 5459969, 5742593, 5939201, 6328321, 6369281, 6553601, 6709249, 6721537, 6807553, 6868993, 6983681, 7045121, 7094273, 7217153, 7426049, 7720961, 7950337, 7954433, 8048641, 8101889, 8237057, 8417281, 8421377, 8491009, 8630273, 8704001, 8839169, 8962049, 9097217, 9240577, 9424897, 9510913, 9535489, 9650177, 9883649, 9895937, 9957377, 10190849, 10211329, 10227713, 10608641, 10616833, 10764289, 11194369, 11329537, 11366401, 11542529, 11554817, 11857921, 12091393, 12255233, 12341249, 12374017, 12435457, 12476417, 12832769, 12914689, 13455361, 13692929, 13742081, 13762561, 13828097, 14254081, 14462977, 14491649, 14786561, 14794753, 14868481, 14897153, 14917633, 14921729, 14979073, 15175681, 15179777, 15347713, 15433729, 15716353, 15904769, 15986689, 16097281, 16273409, 16642049, 16834561, 17817601, 17883137, 17891329, 18923521, 18972673, 19341313, 19685377, 19832833, 19898369, 20021249, 20201473, 20561921, 20832257, 20889601, 20930561, 20979713, 21037057, 21250049, 21405697, 21422081, 21651457, 22282241, 22978561, 23633921, 24158209, 24379393, 24788993, 24895489, 24936449, 24985601, 25133057, 25337857, 25477121, 25747457, 27983873, 28016641, 28106753, 28549121, 28794881, 29065217, 31154177, 31670273, 32120833, 32432129, 32464897, 32555009, 33611777, 33660929, 34398209, 34799617, 34922497, 35602433, 35930113, 35954689, 36864001, 37650433, 37724161, 38207489, 39641089, 42508289, 44261377, 46858241, 47153153, 47472641, 48586753, 49815553, 50700289, 51904513, 52559873, 54091777, 54435841, 54624257, 55263233, 55689217, 56303617, 56934401, 59219969, 61857793, 64356353, 65765377, 67092481, 68960257, 79298561, 113459201, 128581633, 136757249, 137789441, 150077441, 160694273, 162611201, 164331521, 166690817, 172638209, 172736513, 181075969, 183451649, 184238081, 189235201, 189792257, 194838529, 195985409, 196558849, 198000641, 199016449, 199262209, 199376897, 200458241, 202260481, 203784193, 207339521, 208404481, 210927617, 211402753.

Some larger examples :