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cake n.  and  gapful n.

First numbers that belong to both sets:

130, 176, 1160, 2325, 2626, 14235, 18473, 19650, 70376, 73227, 82240, 88642, 117570, 138510, 142976, 152194, 161800, 204264, 247020, 280960, 447720, 487488, 551450, 670080, 1038405, 1072632, 1125370, 1217060, 1313600, 1373904, 1415115, 1521730, 1565832, 1633570, 1656576, 1750760, 1873425, 2028026, 2135680, 2275520, 2333122, 2511744, 2796417, 2862468, 2895880, 3244410, 3317312, 3428700, 4322637, 4455400, 4636608, 4822664, 5013632, 5359838, 6493320, 7085050, 10116738, 10194160, 10587200, 10990215, 11319893, 11403330, 11571432, 11826670, 11912576, 12172788, 12260360, 13067483, 13159290, 13344192, 13624780, 14101120, 14886144, 15086850, 15596490, 15699776, 16012368, 16117480, 17194010, 17749800, 18317440, 18663764, 19488770, 19608576, 20709000, 21211008, 22370049, 22633220, 23300160, 23980075, 24117626, 24533433, 25953928, 26098920, 26980226, 28339040, 28492776, 29113280, 32017154, 34056570, 34404672, 37644930, 42473880, 43486720, 47699080, 48572928, 51030900, 51944858, 54277353, 56922600, 59904832, 61949760, 62468162, 63251175, 64040704, 66448657, 71160258, 75793410, 76386312, 79092651, 85014400, 101994050, 103806690, 104171776, 105640680, 107123328, 108619784, 109373210, 111272000, 113192640, 117099970, 118289459, 119086910, 119486976, 121096200, 122312554, 123127965, 123946992, 125182330, 127259420, 129359360, 131482275, 133197233, 133628290, 134493192, 135361818, 135797530, 136234176, 137110274, 137990120, 140206185, 141996023, 142445850, 143348352, 144709240, 146537118, 146996480, 151643010, 152112976, 152583912, 154002550, 154477376, 156386440, 161227770, 163685460, 165669498, 166168000, 166667501, 170192064, 171208130, 173765970, 174280576, 176349160, 181592090, 184252080, 185323307, 186937920, 188561884, 192387650, 195151790, 195707776, 197942280, 200193728, 201325826, 201893472, 205321769, 206473100, 207050376, 209370240, 217029359, 220626448, 221229800, 227939426, 230412520, 233528960, 234783362, 236673375, 240483776, 246275720, 246924951, 251501544, 256134340, 259478080, 264888384, 270373376, 273144040, 278742024, 283702560, 284415976, 287281600, 290166408, 291616026, 293799493, 301898760, 305628925, 310902592, 317003520, 327871258, 329443776, 331021314, 332603880, 342205312, 361958976, 373825530, 375541632, 391223490, 400121320, 413719440, 418319680, 437061640, 440875968, 456355200, 464228864, 472192576, 476208360, 477215851, 482274626, 491470280, 496629120, 507054850, 513379776, 515499714, 533755300, 538112458, 544692555, 552436872, 560254248, 561377000, 593426304, 625464540, 636394728, 641293184, 656139240, 681388800, 687800815, 691667264, 696845032, 703353497, 713851125, 717814504, 780436098, 788863040, 818834751, 828991464, 846597760, 849556162, 913285128, 924206976, 978495915, 996504576, 1003107560, 1011402225, 1014732852, 1019742490, 1023091392, 1028128480, 1036560320, 1045038135, 1050146944, 1053562050.

Some larger examples :