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plaindrome  and  strong prime

First numbers that belong to both sets:

11, 17, 29, 37, 59, 67, 79, 127, 137, 149, 179, 223, 227, 239, 269, 277, 347, 367, 379, 457, 479, 499, 557, 569, 599, 1117, 1229, 1249, 1277, 1289, 1399, 1447, 1567, 1579, 1667, 1777, 2237, 2267, 2333, 2339, 2347, 2377, 2389, 2459, 2467, 2579, 2789, 2999, 3359, 3389, 3449, 3457, 3467, 3557, 4447, 4799, 4999, 5557, 5569, 5689, 5779, 6689, 6779, 6899, 7789, 8999, 11113, 11117, 11149, 11159, 11239, 11257, 11369, 11467, 11489, 11579, 11677, 11699, 11777, 12227, 12239, 12277, 12377, 12569, 12577, 12689, 12889, 13337, 13367, 13567, 13669, 13679, 13799, 14447, 14479, 14557, 14699, 14779, 15667, 15679, 16699, 17789, 18899, 22229, 22247, 22277, 22367, 22469, 22567, 22669, 22777, 23357, 23369, 23399, 23447, 23557, 23599, 23669, 23789, 24469, 24499, 25577, 25667, 25799, 25889, 26669, 26699, 26777, 27799, 33347, 33377, 33457, 33479, 33577, 33599, 33679, 33889, 34457, 34469, 34499, 34589, 34667, 35569, 36779, 36899, 37799, 37889, 44449, 44579, 44699, 44789, 45667, 46679, 47777, 55579, 55889, 56779, 57899, 58889, 59999, 67777, 77999, 111119, 111227, 111269, 111337, 111467, 111577, 111779, 112237, 112247, 112279, 112289, 112337, 112459, 112559, 112799, 113357, 113489, 113557, 113777, 113899, 114467, 114599, 114689, 115589, 115679, 115777, 116789, 122347, 122399, 122449, 122489, 122557, 122777, 123377, 123449, 123479, 123499, 124447, 124489, 124669, 124699, 124777, 125777, 125789, 133349, 133379, 133447, 133559, 133669, 134489, 134669, 134677, 134699, 134777, 134999, 135559, 135589, 135599, 138889, 144479, 144577, 144589, 144667, 145577, 145679, 145799, 146669, 146677, 155557, 155569, 155579, 155689, 156677, 156899, 157889, 157999, 166667, 166799, 167777, 168899, 178889, 179999, 188999, 199999, 222247, 222269, 222289, 222347, 222379, 223337, 223577, 223667, 223679, 223999, 224669, 224677, 224699, 225569, 225779, 226777, 227789, 233477, 234457, 234499, 234799, 235577, 235889, 236699, 236779, 239999, 244667, 244889, 246889, 246899, 247799, 256799, 266677, 267899.

Some larger examples :