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 |
2 | 0 | 1038 | 2 |
3 | 0 | 540 | 498 | 3 |
4 | 0 | 277 | 0 | 761 | 4 |
5 | 0 | 909 | 5 | 118 | 6 | 5 |
6 | 0 | 540 | 0 | 0 | 0 | 498 | 6 |
7 | 0 | 186 | 181 | 185 | 167 | 157 | 162 | 7 |
8 | 0 | 200 | 0 | 232 | 0 | 77 | 0 | 529 | 8 |
9 | 0 | 186 | 147 | 0 | 242 | 222 | 0 | 112 | 129 | 9 |
10 | 0 | 909 | 0 | 118 | 0 | 0 | 0 | 5 | 0 | 6 | 10 |
11 | 0 | 97 | 110 | 114 | 117 | 108 | 94 | 82 | 103 | 123 | 90 |
A pictorial representation of the table above
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.