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alternating n.  and  Cunningham n.

First numbers that belong to both sets:

3, 5, 7, 8, 9, 10, 50, 63, 65, 101, 127, 129, 143, 145, 290, 323, 325, 783, 785, 901, 1090, 1850, 2303, 2305, 2501, 2703, 2705, 2743, 2745, 3250, 4901, 6723, 6725, 8101, 9410, 12101, 12543, 12545, 16383, 16385, 16901, 21610, 23410, 27890, 32767, 32769, 36101, 41210, 47090, 52901, 58563, 58565, 61010, 72901, 85850, 96101, 127450, 161050, 161052, 165650, 187490, 232323, 232325, 238143, 238145, 238327, 238329, 278783, 278785, 305810, 321490, 363610, 389016, 389018, 412163, 412165, 432963, 432965, 476101, 567010, 656101, 727610, 830583, 830585, 898703, 898705, 1036323, 1036325, 1056783, 1056785, 1098303, 1098305, 1232101, 1236543, 1236545, 1272383, 1272385, 1276901, 1416101, 1612901, 1658943, 1658945, 1876901, 2301290, 2505890, 2569610, 2745650, 3034563, 3034565, 3090563, 3090565, 3276101, 3452163, 3452165, 3496901, 3678723, 3678725, 3810303, 3810305, 4549690, 4765490, 5098563, 5098565, 5234943, 5234945, 5616901, 6165290, 6589490, 6723650, 6967870, 6967872, 6985450, 7236101, 7452901, 7672901, 7896101, 8105410, 8369450, 8743850, 9072143, 9072145, 9290303, 9290305, 9498723, 9498725, 9672101, 9872163, 9872165, 10349090, 10503458, 10543010, 10909810, 12369290, 12581210, 12723490, 12909650, 12981610, 14569490, 16345850, 16785410, 16949690, 18983450, 21252101, 21270543, 21270545, 21418383, 21418385, 21436901, 21696963, 21696965, 23892543, 23892545, 25412183, 25412185, 27478563, 27478565, 29030543, 29030545, 29052101, 30327050, 30769210, 32569850, 34187410, 34727450, 43692101, 45050943, 45050945, 45292901, 45454563, 45454565, 45670563, 45670565, 45832901, 47032163, 47032165, 47472101, 47858723, 47858725, 49252323, 49252325, 49674303, 49674305, 50367410, 50509450, 52521874, 52521876, 54125450, 56145050, 56505290, 61496963, 61496965, 63234303, 63234305, 67076101, 67436943, 67436945, 69056101, 69856163, 69856165, 70509610, 72301010, 72369050, 74701450, 74943650, 78907690, 81216143, 81216145, 85414563, 85414565, 87272963, 87272965, 87834383, 87834385, 90307010, 90383050, 92987450, 96963410, 103836101, 105472901, 109872323, 109872325, 121616783, 121616785, 123432101, 123476543, 123476545, 123832383, 123832385, 123876901, 125216101, 127012901, 127418943, 127418945, 129276901, 145250703, 145250705, 147816963, 147816965, 161036101, 163072901, 165636901, 167650703, 167650705, 181494783, 181494785, 183656703, 183656705, 185232101, 187416101, 189612901, 210163010, 210743290, 230523490, 232349050, 234181810, 252587450, 252905410, 254945090, 270503810, 270701210.

Some larger examples :