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m-pointer prime  and  prime n.

First numbers that belong to both sets:

23, 61, 1123, 1231, 1321, 2111, 2131, 11261, 11621, 12113, 13121, 15121, 19121, 21911, 22511, 27211, 61211, 116113, 131231, 312161, 611113, 1111211, 1111213, 1111361, 1112611, 1123151, 1411411, 1612111, 2111411, 2121131, 3112111, 3116111, 3221111, 4121113, 9111341, 11113321, 11132123, 11264111, 11314111, 11316211, 11361211, 11413111, 11611121, 11922121, 12111163, 12111313, 12111331, 13121117, 14111131, 14111311, 19122211, 23311111, 61114211, 111111163, 111131213, 111219211, 111316111, 111611113, 112121213, 114131141, 115121311, 116112131, 119112113, 121115311, 121151341, 121235111, 131121131, 131211181, 131261111, 136211111, 163111111, 211121213, 213111131, 311221111, 316111111, 411221311, 1111126111, 1111131821, 1111211113, 1111211161, 1111214123, 1111231111, 1111231151, 1112111911, 1112312321, 1113114421, 1116111151, 1116124111, 1117122113, 1119413111, 1121111531, 1121611411, 1123131511, 1135411111, 1161111113, 1211311111, 1211711131, 1212232111, 1231112111, 1232111161, 1312111111, 1312213121, 1321141121, 2112116131, 2112311141, 2115911111, 2191111121, 2232113111, 3111111121, 4111231211, 4111311131, 4223111111, 6121118111, 11111112191, 11111113261, 11111211491, 11111219111, 11111231321, 11111383111, 11112111191, 11112119111, 11112311113, 11112313111, 11112911111, 11113213211, 11121142231, 11121231151, 11126111111, 11126111131, 11126221111, 11141112431, 11151122119, 11153112121, 11161111211, 11161211111, 11161312111, 11211121123, 11212111213, 11213112611, 11231161111, 11311111261, 11311112611, 11311312141, 11321112113, 11611411121, 12111121223, 12112111331, 12113312111, 12121432111, 12123116111, 12132131111, 12141111611, 12211113221, 12211212161, 12212111113, 12219121111, 12331112111, 12641111111, 13113211411, 13131111211, 13218112111, 13411112311, 14111111153, 16111121113, 16111121311, 16111211131, 16115111113, 16212111121, 17121311111, 21111111143, 21111113251, 21111116311, 21111321611, 21112321211, 21113111213, 21113131141, 21131311511, 21132113111, 21161111141, 21211211123, 21212132111, 21213111131, 22121121311, 22122111311, 22123111121, 23113111211, 23123111311, 31111221611, 31121312111, 31211121131, 31411113121, 32111121113, 41134111111, 41141311121, 43112311111, 51211113131, 53122111111, 61511111111, 73111111211, 91141111211, 111111115123, 111111116117, 111111123121, 111111133121, 111111212131, 111111322111, 111111491111, 111111811123, 111113132111, 111113611211, 111114111133, 111114132121, 111116121131, 111121112213, 111121115311, 111121311151, 111122113241, 111129112111, 111131212111, 111131222131, 111132118121, 111132221311, 111141312211, 111152111311, 111161111311, 111161125111, 111211231111, 111211921211, 111212313211, 111214113211, 111221111621, 111221131111, 111242131111, 111311112113, 111311511121, 111321121111, 111331211111, 111621111131, 112112111621, 112131113221, 112131121231, 112131161111, 112211211113, 112311115121, 112511311111, 112611111331, 112911111113, 113112421111, 113114113121, 113121111613, 113211132211, 113221111111, 113221121311, 113612111111, 114111212113, 114121311131, 114211131121, 114622111111, 121113412111, 121121112191, 121121161121, 121151111131, 121152111131, 121191111211, 121311151111, 123111111181, 123112111411, 123114131111, 126111121211, 131111211151, 131112112231, 131113221121, 131115211111, 131121811111, 131151211111, 131213111111, 131222111131, 131231111111, 131311711211, 133111111121.

Some larger examples :