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hexagonal n.  and  interprime n.

First numbers that belong to both sets:

6, 15, 45, 120, 231, 780, 861, 1431, 3321, 3828, 4005, 6903, 7140, 7381, 7875, 11175, 11781, 13695, 14706, 17205, 30135, 32640, 39621, 40755, 44253, 46056, 49770, 51681, 58311, 60378, 69006, 71253, 80601, 84666, 88831, 89676, 90525, 92235, 94830, 97461, 108345, 139128, 147696, 158766, 172578, 177310, 190653, 201930, 205761, 208335, 214840, 216153, 229503, 267546, 269011, 271953, 280875, 288420, 303810, 321201, 360825, 378015, 475800, 477753, 483636, 489555, 499500, 511566, 546535, 574056, 606651, 615495, 635628, 679195, 714610, 726615, 748476, 753378, 763230, 768180, 783126, 793170, 795691, 808356, 836571, 846951, 862641, 870540, 959805, 962578, 982101, 1007490, 1016025, 1044735, 1097421, 1115271, 1121253, 1142316, 1154440, 1166628, 1169685, 1188111, 1197378, 1203576, 1206681, 1228528, 1234806, 1269621, 1350546, 1396956, 1400301, 1433971, 1519896, 1540890, 1619100, 1626306, 1662576, 1666225, 1673535, 1714026, 1758750, 1773786, 1804050, 1819278, 1853775, 1979055, 1983036, 2007006, 2051325, 2141415, 2153850, 2174655, 2212356, 2263128, 2280180, 2301585, 2323090, 2388205, 2418900, 2449791, 2507680, 2570778, 2620905, 2676141, 2708628, 2736630, 2755378, 2859636, 2864421, 2874003, 2951235, 3039345, 3044278, 3083886, 3173940, 3219453, 3234696, 3244878, 3332071, 3357936, 3368310, 3383901, 3430890, 3499335, 3504628, 3573801, 3719628, 3873936, 3890655, 3907410, 3974790, 3991725, 4054128, 4059825, 4174605, 4197753, 4232595, 4279275, 4308580, 4320330, 4355676, 4528545, 4546620, 4607130, 4772505, 4828278, 5073705, 5221296, 5338278, 5542785, 5596185, 5764710, 5839653, 5853331, 6130251, 6137256, 6158295, 6200481, 6235746, 6242811, 6306576, 6528691, 6601161, 6695970, 6909903, 6932226, 6999411, 7051890, 7066920, 7134753, 7142310, 7157436, 7172578, 7279020, 7301931, 7370880, 7409325, 7548555, 7556328, 7587460, 7649916, 7910253, 7958055, 8134561, 8296701, 8353828, 8386560, 8419356, 8435778, 8551180, 8567730, 8592585, 8986680, 9046131, 9054640, 9097245, 9191328, 9217071, 9372285, 9528795, 9616305, 9633855, 9774831, 9792525, 9836830, 9881235, 9952491, 9979278, 10059855, 10131751, 10176816, 10249128, 10258185, 10285380, 10385403, 10440165, 10467600, 10669890, 10679131, 10836840, 10939503, 11005086, 11288376, 11508003, 11536806, 11546415, 11556028, 11575266, 11710380, 11768526, 11807370, 12179580, 12238878, 12268581, 12427605, 12587653, 12637878, 12870201, 12961686, 13053495.

Some larger examples :