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Proth n.  and  sphenic n.

First numbers that belong to both sets:

385, 609, 705, 897, 1281, 1729, 2945, 3009, 3201, 3585, 3905, 4865, 5505, 5889, 6657, 6785, 7553, 7809, 8449, 8961, 11137, 11265, 11649, 12545, 12673, 12801, 14465, 14721, 15617, 15745, 16385, 17409, 18177, 19201, 22017, 25345, 27393, 29185, 29697, 29953, 31233, 31745, 32513, 33537, 34305, 34561, 35585, 35841, 36865, 38145, 38657, 39169, 40449, 40705, 41217, 42497, 43265, 43521, 44545, 45057, 46081, 48129, 49153, 50945, 51969, 52481, 54273, 55041, 56833, 57345, 58881, 59649, 60673, 60929, 61185, 61953, 64001, 66561, 67585, 68609, 71169, 74753, 82433, 82945, 84993, 91137, 94209, 94721, 99841, 100353, 104961, 108545, 110081, 111617, 113665, 114177, 118785, 121345, 121857, 125953, 126465, 126977, 128513, 132097, 135169, 136705, 137217, 139777, 140289, 144385, 148993, 149505, 150017, 152065, 154113, 158721, 159745, 164353, 168961, 169473, 171521, 172545, 173569, 175105, 178177, 181761, 182273, 182785, 185345, 187905, 190465, 198145, 200193, 200705, 201729, 202241, 203777, 204289, 204801, 205313, 206337, 207361, 208385, 214529, 215553, 216065, 221185, 223745, 225793, 227841, 229377, 232449, 237057, 241665, 243713, 246273, 250369, 250881, 251905, 252417, 252929, 255489, 256513, 259585, 260097, 269313, 278529, 280577, 284673, 287745, 288769, 291841, 293889, 297985, 302081, 304129, 313345, 315393, 316417, 317441, 321537, 325633, 335873, 339969, 340993, 343041, 347137, 348161, 349185, 352257, 358401, 360449, 367617, 368641, 369665, 374785, 375809, 376833, 381953, 397313, 408577, 413697, 415745, 420865, 422913, 431105, 432129, 435201, 441345, 445441, 450561, 451585, 453633, 460801, 466945, 477185, 481281, 483329, 485377, 490497, 491521, 492545, 496641, 504833, 505857, 506881, 516097, 517121, 518145, 523265, 526337, 527361, 535553, 543745, 547841, 553985, 556033, 561153, 563201, 569345, 570369, 573441, 574465, 579585, 582657, 584705, 586753, 588801, 591873, 595969, 598017, 600065, 605185, 607233, 608257, 618497, 619521, 620545, 621569, 635905, 644097, 652289, 655361, 656385, 657409, 661505, 663553, 669697, 672769, 678913, 684033, 686081, 688129, 691201, 693249, 699393, 701441, 702465, 707585, 710657.

Some larger examples :