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de Polignac n.  and  heptagonal n.

First numbers that belong to both sets:

1, 9517, 96727, 101707, 135373, 191407, 221563, 258727, 354757, 364237, 395413, 487747, 523723, 733597, 763693, 869365, 1039095, 1081423, 1121245, 1131313, 1269853, 1754353, 2250079, 2409337, 2796823, 3100819, 3123133, 3207523, 3344887, 3467443, 3923143, 3979717, 4106887, 4151869, 4340833, 4745143, 5637757, 5811775, 5972971, 6286111, 6664273, 7070287, 7163083, 7196977, 7402021, 7445101, 7479655, 7679893, 8034433, 8079313, 8396973, 8433667, 9338323, 9581473, 10017007, 10167697, 10258651, 10906669, 11585293, 11682367, 11845057, 12240103, 12295483, 12912913, 13697191, 13885087, 13944067, 14264719, 14661577, 14758605, 14843767, 14990329, 15335107, 15897427, 16036423, 16087117, 17327773, 17393653, 18737503, 18847171, 18943393, 19275157, 19778203, 20530591, 20702893, 20991663, 21093205, 21501823, 21796093, 21929167, 22345765, 23297443, 23373823, 24597217, 25308037, 26776413, 27187063, 28252567, 29389959, 29458573, 29802843, 30567277, 30935533, 31625287, 31981957, 32268733, 32629003, 32791777, 33063967, 33082153, 33300775, 33520117, 33666745, 34237951, 34460353, 34553233, 35206893, 35281987, 36647359, 36896647, 37108243, 37882783, 38351347, 39455863, 39535357, 40254403, 42059857, 43382641, 46154077, 46261557, 47473873, 48063985, 48437407, 48944925, 49655437, 49766917, 50303761, 51001447, 51817693, 52365457, 53400277, 53654773, 53863447, 54026029, 54795787, 55053583, 55641733, 55830601, 57042157, 57544813, 58508353, 59455507, 59577487, 61943743, 62242765, 63925537, 65682001, 66195571, 66504673, 66607867, 67644207, 68688547, 70270057, 70987609, 71094223, 71521479, 72703033, 73894267, 73921453, 74220829, 74575417, 76112533, 76997925, 77192287, 78307627, 79205659, 79995637, 80137117, 80562307, 81130977, 81615919, 82274317, 82303003, 82877773, 83772619, 84614083, 86927677, 87075157, 88259497, 90381403, 90411469, 91830211, 92924377, 93565633, 94393345, 98022217, 99026649.