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frugal n.  and  Proth n.

First numbers that belong to both sets:

10625, 14337, 16129, 66049, 90625, 134657, 138753, 261121, 263169, 1009665, 1050625, 1507329, 1540097, 1615873, 2502657, 3020801, 3995649, 4190209, 4198401, 4890625, 5488641, 6533121, 8474625, 10465281, 10522625, 10539009, 10903553, 11378689, 11460609, 12058625, 12570625, 13557761, 15130625, 16367617, 16785409, 17432577, 19357697, 19537921, 20193281, 20250625, 23404545, 25649153, 27107329, 29376513, 30490625, 35348481, 36036609, 36634625, 41320449, 45850625, 47210497, 50970625, 53264385, 55705601, 58122241, 65208321, 67092481, 67125249, 68321281, 81690625, 83230721, 89096193, 104185857, 105742337, 106266625, 107560961, 108789761, 109527041, 110100481, 112410625, 112803841, 116932609, 120946689, 122650625, 124698625, 124928001, 125779969, 129286145, 130154497, 132890625, 134135809, 134381569, 136871937, 140853249, 144834561, 148815873, 151928833, 156778497, 156860417, 160759809, 164741121, 172015617, 172703745, 173719553, 175095809, 180338689, 184090625, 188874753, 189202433, 195821569, 196591617, 200572929, 200835073, 208535553, 208551937, 218677249, 224296961, 239190017, 244367361, 245530625, 246857729, 252395521, 254017537, 258342913, 268255233, 270827521, 280199169, 286490625, 296189953, 304087041, 325451777, 327450625, 327974913, 335282177, 347930625, 361496577, 370409473, 375750657, 388890625, 399638529, 423526401, 447414273, 449085441, 491290625, 495190017, 527761409, 552730625, 590741505, 614629377, 662405121, 665288705, 696090625, 725581825, 734068737, 758022145, 790265857, 798490625, 805732353, 837910529, 842465281, 870285313, 874971137, 877395969, 900890625, 921141249, 954007553, 999817217.