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m-pointer primes
A prime number p such that the next prime number can be obtained adding p to its product of digits. more

The first 600 m-pointer primes :
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, 141111123211, 141113221111, 142311111211, 151121161111, 161181111211, 162122111111, 211111123141, 211111231411, 211111325111, 211112111311, 211115131111, 211131211141, 211131611111, 211143115111, 211311111821, 211622111111, 212121114311, 212411131111, 221111114131, 221113131121, 311113112111, 311115112111, 311162111111, 311211112151, 311211211321, 311215511111, 311411121121, 312111211141, 312211111111, 321111111131, 321111211111, 411112112113, 411131122111, 411511213111, 413212121111, 413311111121, 421111311121, 441111111131, 511112112131, 511112131111, 511211311111, 611121111143, 611153111111, 613511111111, 711611111111, 1111111111219, 1111111112611, 1111111241113, 1111111382111, 1111111615121, 1111111652111, 1111111813121, 1111112111191, 1111112111413, 1111112611111, 1111113111211, 1111113112121, 1111116121411, 1111116811111, 1111117131211, 1111117219111, 1111119112211, 1111121123221, 1111121153111, 1111121211311, 1111121811113, 1111124112311, 1111131126113, 1111131281111, 1111131521111, 1111132111231, 1111132132111, 1111191118111, 1111211119121, 1111211123117, 1111212111133, 1111212313121, 1111212314111, 1111231151111, 1111311221221, 1111313211121, 1111362112111, 1111371121111, 1111511312111, 1111611111731, 1112111112361, 1112111118113, 1112111231111, 1112113222111, 1112113312111, 1112121221231, 1112611221121, 1112613112111, 1113111112111, 1113112711111, 1113114111113, 1113121112111, 1113121121113, 1113211122131, 1115111312141, 1115112111131, 1115116112111, 1115126111111, 1116111116111, 1116121113113, 1116161111111, 1116311212111, 1117111111231, 1118121111113, 1118711111113, 1119122111111, 1121111111311, 1121111315111, 1121112111311, 1121161212121, 1121211121231, 1121221321111, 1121312113111, 1121313112111, 1121371111111, 1122111231211, 1122122111113, 1122211112311, 1131111122111, 1131112111211, 1131141113111, 1131211111211, 1131212111221, 1131213211211, 1132221111311, 1133212112111, 1134111511111, 1141112611211, 1141311211211, 1211121111131, 1211132111311, 1211143112111, 1211153111111, 1211211111311, 1211211161213, 1211212211311, 1211222111611, 1211312141111, 1211323111211, 1211331111121, 1212111131111, 1212112112131, 1212121211131, 1213211111113, 1215111112231, 1221113112311, 1221313111111, 1231111111123, 1231113112111, 1231121111131, 1232112111211, 1261111311121, 1311111125111, 1311111512111, 1311113121211, 1311114511121, 1312111112141, 1313111121211, 1321123111211, 1321611112111, 1411123211111, 1411132115111, 1411211411131, 1412311112111, 1511111191211, 1611211111141, 1612311111211, 1811111211311, 1811123111111, 1911113211121, 2111111116111, 2111111119121, 2111111161111, 2111111211311, 2111111312311, 2111111321131, 2111116212211, 2111121112123, 2111171113111, 2111191111111, 2111231122111, 2111311121131, 2112131121211, 2113111111213, 2118111311111, 2121111163111, 2123121121111, 2131111111121, 2131222111111, 2132111111131, 2132113111111, 2141111133311, 2161231111111, 2211111611311, 2211431111111, 2221111111123, 2221121111113, 2231111111111, 2312151111211, 2411111131211, 3111111112111, 3111111121121, 3111121171111, 3111123111121, 3111131111411, 3111221112121, 3111411211121, 3112111112311, 3112121112131, 3114111111113, 3121111141411, 3121111151141, 3121221211211, 3141112211113, 3211223111111, 3431111111111, 4111118131111, 4111131111611, 4111212311111, 4111311421111, 4121231111111, 4311111116111, 5121131111111, 6111111142121, 11111111211413, 11111111213611, 11111112122113, 11111112131111, 11111112163111, 11111116111321, 11111119221211, 11111123121113, 11111125132111, 11111126121211, 11111131219211, 11111131232111, 11111133212111, 11111142311221, 11111153141111, 11111155113211, 11111192111141, 11111251111321, 11111411141113, 11111611111171, 11111911111811, 11112111111631, 11112111112133, 11112111216121, 11112111531113, 11112211211161, 11112216113111, 11112221411113, 11113111181411, 11113111411111, 11113111411411, 11113111612111, 11113311111211, 11113711111121, 11114111513111, 11114141112311, 11114311311121, 11121124131211, 11121191311121, 11121311131211, 11121321116111, 11121611111131, 11122111131151, 11123111121113, 11131112111123, 11131114132111, 11131121111711, 11131211113111, 11131211131111, 11131221141211, 11131231111111, 11136121111111, 11141111213311, 11141121314111, 11142211113211, 11171311111241, 11211111111611, 11211111113113, 11211112113241, 11211113125111, 11211121121231, 11211121311311, 11211161111131, 11211212113141, 11211212411131, 11211221111113, 11211313121111, 11212126111111, 11212132121111, 11212133111111, 11213111122141, 11213113114111, 11215211211311, 11223121111211, 11226111111121, 11231117111111, 11231421111121, 11251113122111, 11311111126121, 11311111144121, 11311152211111, 11311212111131, 11311232311111, 11411211133111, 11511611114111, 11512111113131, 11512112131111, 11611111121111, 11611111211411, 11611111312111, 11611121111131.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1038 values, from 23 to 911311112211121).

n\r 0  1 
201038 2 
30540498 3 
402770761 4 
5090951186 5 
60540000498 6 
70186181185167157162 7 
8020002320770529 8 
9018614702422220112129 9 
1009090118000506 10 
11097110114117108948210312390

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.