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m-pointer primes
A prime number  $p$  is called m-pointer if the next prime number can be obtained adding  $p$  to its product of digits (here the 'm' stands for multiplicative).

For example, 1231 is a m-pointer prime since the next prime is equal to 1231 + 1 ⋅ 2 ⋅ 3 ⋅ 1= 1237.

The first m-pointer primes are 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 more terms

M-pointer primes can also be... (you may click on names or numbers and on + to get more values)

a-pointer 15121 aban 23 61 alternating 23 61 amenable 61 1321 11261 11621 12113 13121 15121 19121 116113 312161 + 119112113 121151341 131211181 211121213 apocalyptic 1321 2131 11261 11621 12113 13121 15121 19121 21911 22511 27211 arithmetic 23 61 1123 1231 1321 2111 2131 11261 11621 12113 + 3116111 3221111 4121113 9111341 balanced p. 1123 21911 3116111 11413111 12111331 14111311 316111111 1111131821 c.decagonal 61 c.square 61 Chen 23 2111 11261 11621 12113 13121 19121 21911 22511 61211 + 2121131 9111341 11922121 14111131 congruent 23 61 1231 1321 2111 11261 11621 21911 22511 131231 + 3112111 3116111 3221111 9111341 Curzon 1111361 9111341 cyclic 23 61 1123 1231 1321 2111 2131 11261 11621 12113 + 3116111 3221111 4121113 9111341 d-powerful 2131 de Polignac 15121 61211 1111211 13121117 61114211 deficient 23 61 1123 1231 1321 2111 2131 11261 11621 12113 + 3116111 3221111 4121113 9111341 dig.balanced 15121 2111411 2121131 11113321 11264111 13121117 61114211 136211111 163111111 economical 23 61 1123 1231 1321 2111 2131 11261 11621 12113 + 13121117 14111131 14111311 19122211 emirp 1231 1321 11621 12113 3112111 9111341 11264111 11413111 12111163 12111313 + 111111163 121115311 131261111 163111111 equidigital 23 61 1123 1231 1321 2111 2131 11261 11621 12113 + 13121117 14111131 14111311 19122211 esthetic 23 evil 23 11621 12113 13121 22511 27211 116113 312161 1112611 1123151 + 163111111 213111131 316111111 411221311 fibodiv 61 good prime 61211 312161 happy 23 2111 15121 116113 611113 1111211 hex 61 Hogben 1123 iban 23 1123 1321 2111 27211 inconsummate 11261 22511 61211 junction 2111 13121 15121 19121 22511 27211 61211 116113 611113 1112611 + 12111313 13121117 14111311 23311111 katadrome 61 Kynea 23 lonely 23 lucky 1123 1231 15121 4121113 magnanimous 23 61 metadrome 23 modest 23 2111 3221111 23311111 136211111 163111111 316111111 1312111111 nialpdrome 61 2111 3221111 oban 23 odious 61 1123 1231 1321 2111 2131 11261 15121 19121 21911 + 131211181 131261111 211121213 311221111 Ormiston 116113 611113 12111313 12111331 14111131 111131213 111611113 1161111113 palindromic 131121131 1116111116111 palprime 131121131 1116111116111 panconsummate 23 61 pernicious 61 1123 1231 1321 2111 2131 11261 15121 19121 61211 + 2111411 2121131 3116111 9111341 plaindrome 23 1123 prime 23 61 1123 1231 1321 2111 2131 11261 11621 12113 + 611121111143 611153111111 613511111111 711611111111 repfigit 61 repunit 1123 self 12113 1111213 11264111 61114211 121235111 131261111 136211111 strong prime 2111 13121 15121 19121 61211 312161 1111211 1112611 1411411 1612111 + 11113321 11611121 14111131 23311111 super-d 1123 1231 2131 19121 131231 1123151 2121131 truncatable prime 23 12113 twin 61 1231 1321 2111 2131 611113 1111211 1111213 4121113 11314111 + 12111313 111219211 111316111 311221111 uban 23 61 Ulam 15121 19121 1111211 1111213 weak prime 23 61 1231 1321 2131 11261 11621 12113 22511 27211 + 12111313 13121117 19122211 61114211 Woodall 23