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Honaker prime  and  inconsummate n.

First numbers that belong to both sets:

1049, 1571, 2591, 3137, 4091, 5507, 5927, 13037, 13217, 14843, 15671, 16661, 21617, 26309, 29101, 29243, 31391, 31517, 34259, 34703, 35027, 35327, 35537, 37019, 37049, 40637, 41141, 42323, 42737, 46237, 51421, 51473, 54617, 55541, 59183, 60251, 62627, 65843, 68147, 69257, 70379, 72701, 74231, 77237, 77611, 78437, 80657, 88007, 90647, 103619, 106031, 109133, 110273, 111491, 114809, 115571, 115601, 115631, 115853, 123701, 125627, 136217, 136337, 139703, 142403, 143357, 145121, 145451, 150287, 151337, 152003, 157211, 157307, 173207, 174767, 177007, 180281, 182417, 184187, 185519, 187721, 191627, 192461, 193163, 200237, 201809, 203417, 206951, 208457, 214517, 216917, 217337, 219911, 220469, 223637, 228251, 233117, 234317, 235541, 244313, 247913, 249233, 249329, 250037, 250307, 251501, 256121, 258233, 261071, 261281, 261407, 267017, 270167, 273047, 273233, 273323, 273521, 274163, 274361, 275711, 279431, 280037, 280253, 280703, 281243, 281273, 281807, 282143, 283097, 283139, 284237, 284267, 285251, 285473, 286721, 291701, 293081, 295033, 298223, 300137, 300647, 304481, 306473, 306503, 306727, 307169, 309137, 309539, 310169, 311677, 313637, 313727, 314357, 314651, 314707, 314747, 316259, 317327, 317831, 320339, 320669, 320699, 321289, 325301, 325517, 325643, 325723, 329503, 330917, 333383, 333911, 334877, 335591, 336437, 340643, 341057, 343391, 343667, 345881, 347717, 348149, 356333, 357611, 360167, 360287, 360461, 360821, 361241, 361337, 362717, 363059, 365201, 366001, 370103, 374501, 375113, 375341, 382163, 383723, 385193, 390281, 391403, 393611, 402443, 402803, 405917, 408263, 409217, 418031, 422363, 422393, 423281, 423587, 424727, 431153, 431447, 432251, 432557, 433703, 436181, 436529, 444641, 448373, 453737, 462337, 462983, 465317, 466139, 469241, 470837, 472391, 472817, 473477, 476981, 479861, 482231, 491591, 491651, 492911, 501037, 501707, 501947, 502517, 503621, 505607, 513683, 513917, 525017, 540167, 547061, 549431, 550007, 551363, 551813, 555143, 556043, 562181, 565391, 568109, 570091, 571477, 574031, 576701, 580607, 581177, 584261, 584531, 590027, 590609, 591431, 591827, 595627, 595817, 596507, 598307, 600307, 600659, 600881, 600893, 601451.

Some larger examples :