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congruent n.  and  Cunningham n.

First numbers that belong to both sets:

5, 7, 15, 24, 28, 31, 37, 63, 65, 80, 101, 120, 124, 126, 127, 143, 145, 197, 215, 224, 226, 244, 255, 257, 323, 325, 342, 344, 399, 440, 442, 485, 511, 575, 624, 677, 728, 783, 840, 901, 960, 999, 1023, 1025, 1155, 1157, 1224, 1295, 1330, 1332, 1368, 1443, 1445, 1520, 1599, 1727, 1765, 1848, 1935, 2047, 2117, 2196, 2198, 2208, 2303, 2400, 2501, 2703, 2743, 2808, 2810, 2917, 3024, 3026, 3124, 3126, 3135, 3365, 3374, 3480, 3599, 3601, 3845, 3968, 4095, 4357, 4488, 4490, 4623, 4760, 4899, 4901, 4914, 5183, 5185, 5328, 5475, 5477, 5624, 5775, 5831, 5928, 6083, 6085, 6240, 6242, 6399, 6560, 6725, 7055, 7224, 7395, 7397, 7743, 7775, 7920, 7999, 8001, 8101, 8191, 8280, 8463, 8835, 8837, 9024, 9215, 9262, 9605, 9999, 10001, 10200, 10405, 10608, 10647, 10649, 10815, 11024, 11237, 11448, 11663, 11880, 12101, 12166, 12543, 12545, 12768, 12997, 13455, 13688, 13823, 13925, 14160, 14399, 14640, 14885, 15128, 15375, 15624, 15626, 15877, 16128, 16383, 16385, 16640, 16806, 16901, 17160, 17423, 17575, 17688, 17955, 17957, 18495, 18497, 18768, 19045, 19320, 19599, 19684, 20165, 20448, 20735, 21026, 21317, 21903, 21951, 22200, 22501, 23103, 23408, 23717, 24024, 24335, 24390, 24965, 25280, 25599, 25601, 25920, 26243, 26245, 26895, 26999, 27224, 27555, 27557, 27888, 28223, 28560, 28901, 29240, 29583, 29585, 29790, 29792, 30277, 30975, 30977, 31328, 31685, 32040, 32399, 32760, 32767, 33125, 33488, 33855, 34224, 34597, 34968, 35343, 35345, 35936, 36101, 36863, 36865, 37248, 37637, 38024, 38415, 38808, 39205, 39303, 39999, 40001, 40400, 40805, 41208, 41210, 41615, 42437, 42848, 42876, 43263, 43680, 44101, 44943, 44945, 45368, 45370, 45797, 46655, 46657, 47088, 47090, 47523, 47525, 47960, 48399, 49285, 49728, 50175, 50177, 50624, 50626, 50652, 50654, 51077, 51983, 52440.

Some larger examples :