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gapful n.  and  hoax n.

First numbers that belong to both sets:

160, 660, 1111, 1344, 1484, 1550, 1600, 1650, 1665, 1752, 1812, 2064, 2106, 2236, 2275, 2325, 2500, 2541, 2625, 2640, 2751, 2760, 2960, 3268, 3390, 3552, 3630, 3712, 3930, 4160, 4275, 4905, 4920, 5150, 6060, 6175, 6600, 6900, 6960, 8415, 8720, 9180, 9540, 9630, 10188, 10240, 10278, 10392, 10550, 10560, 10780, 11232, 11712, 11774, 11775, 12192, 12195, 12256, 12552, 12612, 12650, 12780, 12795, 12825, 12880, 12937, 12939, 12964, 12972, 13296, 13428, 13518, 13520, 13560, 13616, 13761, 13770, 13780, 13815, 14084, 14472, 14475, 14480, 14520, 14644, 14712, 14715, 14980, 15150, 15500, 15720, 15764, 15780, 15950, 16000, 16125, 16245, 16425, 16500, 16653, 16752, 16770, 16850, 16860, 16890, 17050, 17145, 17150, 17164, 17175, 17187, 17250, 17616, 17835, 17865, 17880, 17936, 18016, 18018, 18069, 18207, 18284, 18360, 18460, 18564, 18612, 18650, 18735, 18844, 18850, 19080, 19170, 19260, 19550, 19590, 19920, 20660, 21375, 21504, 21625, 21651, 21680, 21775, 22352, 22386, 22460, 22475, 22491, 22516, 22563, 22708, 22960, 23075, 23725, 23784, 23809, 23920, 23925, 24076, 24240, 24260, 24264, 24403, 24480, 24525, 24560, 24620, 24640, 24668, 24980, 25000, 25275, 25508, 25560, 25675, 26060, 26348, 26400, 26419, 26420, 26425, 26580, 26709, 26980, 27188, 27321, 27475, 27540, 27600, 27660, 27748, 27820, 27840, 28104, 28182, 28475, 28525, 28580, 28620, 28680, 28860, 28925, 28940, 29008, 29325, 29600, 29612, 29625, 29660, 30390, 30464, 30750, 30930, 31675, 31740, 31950, 32528, 32880, 32908, 33090, 33336, 33363, 33775, 33864, 33900, 34048, 34112, 34359, 34544, 35187, 35188, 35490, 35790, 35805, 36015, 36036, 36244, 36270, 36300, 36720, 36870, 37320, 38160, 38250, 38340, 38460, 38490, 38520, 39025, 39030, 39300, 39368, 39450, 39570, 39585, 39711, 40725, 41216, 41445, 41600, 41625, 41840, 42075, 42240, 42435, 42640, 42817, 43248, 43280, 43287, 43501, 43680, 44145, 44160, 44480, 44595, 44600, 44763, 44775, 45405.

Some larger examples :