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cake n.  and  Harshad n.

First numbers that belong to both sets:

1, 2, 4, 8, 42, 378, 576, 1160, 2952, 4992, 9920, 10701, 12384, 13288, 14235, 22152, 32568, 50184, 54810, 59712, 73227, 82240, 109824, 125672, 138510, 156948, 204264, 240578, 247020, 310248, 401130, 447720, 562626, 893376, 956040, 1021568, 1125370, 1656576, 2001690, 2054592, 2362284, 2481362, 2511744, 2862468, 2895880, 3101826, 3244410, 3354169, 3940224, 4917530, 5062149, 5160120, 5564644, 5616648, 6436028, 6550951, 7085050, 7711680, 8789376, 9145651, 9437505, 9585732, 9886826, 9963072, 10116738, 10271976, 10990215, 11403330, 11571432, 11912576, 12085634, 12172788, 12260360, 12885152, 13159290, 14101120, 14886144, 15596490, 17749800, 19014426, 19849659, 20214976, 20709000, 22633220, 23300160, 26098920, 26390522, 28957318, 29901495, 30220722, 31028712, 31355898, 32351240, 32519151, 33710824, 34056570, 34931260, 37275264, 38017032, 39338688, 41082504, 41279383, 41476890, 41873792, 42473880, 45140544, 45560450, 51030900, 52175040, 52637442, 54752076, 56192837, 56922600, 57658654, 60411778, 60921576, 61434234, 64305333, 65908430, 68639085, 76386312, 78485808, 81241224, 84695598, 85975604, 86944026, 87593664, 91224588, 91559560, 101274944, 102354876, 105640680, 111272000, 113192640, 118289459, 126010432, 130630344, 134493192, 144709240, 149772826, 150705984, 152583912, 156866151, 161227770, 173765970, 178434048, 184252080, 186937920, 200193728, 201893472, 223654208, 225484092, 227323930, 235412244, 242404218, 251501544, 256134340, 264208022, 266252610, 269683650, 275933568, 278742024, 284415976, 290166408, 292342642, 298199070, 313181180, 331021314, 338182572, 343823298, 351167364, 361958976, 365323400, 375541632, 385074144, 418319680, 421096124, 432324750, 433269376, 456355200, 460280808, 464228864, 466211330, 476208360, 483290652, 491470280, 496629120, 499741684, 500782088, 513379776, 520825074, 530502912, 539205440, 544692555, 548002624, 560254248, 567013265, 593426304, 596930776, 601624830, 602802176, 606343428, 625464540, 656139240, 678835194, 707277960, 717814504, 723121792, 724452689, 737851584, 755515776, 774851610, 803042370, 814507394, 817390600, 839231820, 932061376, 935215554, 963923976, 976869608, 978495915, 1029811176, 1069021668, 1100393127, 1107446901, 1150403465, 1163138432, 1177807360, 1222553088, 1228224712, 1258769824, 1274230476, 1331336000, 1335336002, 1355456232, 1361531470, 1363560576, 1373736351, 1404567176, 1435855876, 1448500317, 1463345728, 1491184227, 1530309510, 1567884032, 1585776960, 1603805504, 1606068633, 1631103720, 1635683322, 1672629114, 1677286080, 1681951682, 1741003632, 1791662464, 1840813704, 1908582390, 1941796584, 1977994242, 1993644224, 2012006610, 2022549848, 2025191400, 2078506560, 2086583644, 2121826176, 2201869395, 2244049640, 2258228505, 2272466970, 2283900738, 2359156360, 2388535290, 2394440192, 2430074088, 2442030144, 2448022850, 2478133480, 2490246528, 2523760800.

Some larger examples :