The sequence starts 2, 3, 11, 23, 29, 61, 19, 113, 157, 127, 103, ... These are the red endings of the following (second column) triangular array. The first column numbers are the indices. The third column numbers are the sums of the digits of the second column integers.
There are several constraints imposed in creating our sequence. Each successive term must be a distinct prime. It must be the smallest such prime that allows the following: At index k, take the final k digits of the sequence. The first of those final k digits must not be zero in order that we may have the concatenation of those digits be a k-digit prime (these are our middle numbers). Finally, the sum of those k digits (the third column) must also be a prime.
1 2 2
2 23 5
3 311 5
4 1123 7
5 12329 17
6 232961 23
7 3296119 31
8 96119113 31
9 119113157 29
10 9113157127 37
11 13157127103 31
12 315712710343 37
13 5712710343149 47
14 71271034314989 59
15 271034314989701 59
16 7103431498970113 61
17 10343149897011341 59
18 343149897011341751 71
19 3149897011341751379 83
20 49897011341751379499 101
21 897011341751379499373 101
22 9701134175137949937397 109
23 11341751379499373971223 101
24 341751379499373971223293 113
25 1751379499373971223293601 113
26 13794993739712232936011471 113
27 949937397122329360114711303 109
28 9937397122329360114711303223 103
29 73971223293601147113032231297 101
30 712232936011471130322312973547 101
31 2232936011471130322312973547619 109
32 32936011471130322312973547619769 127
33 936011471130322312973547619769683 139
34 6011471130322312973547619769683433 137
35 11471130322312973547619769683433503 139
36 471130322312973547619769683433503563 151
37 7113032231297354761976968343350356337 157
38 13032231297354761976968343350356337239 163
39 322312973547619769683433503563372395333 173
40 2312973547619769683433503563372395333337 181
41 12973547619769683433503563372395333337311 181
42 297354761976968343350356337239533333731147 191
43 9735476197696834335035633723953333373114771 197
44 35476197696834335035633723953333373114771673 197
45 761976968343350356337239533333731147716731889 211
46 9769683433503563372395333337311477167318895801 211
47 69683433503563372395333337311477167318895801211 199
48 683433503563372395333337311477167318895801211409 197
49 3433503563372395333337311477167318895801211409277 199
50 33503563372395333337311477167318895801211409277313 199
51 356337239533333731147716731889580121140927731311003 193
52 3372395333337311477167318895801211409277313110031607 193
53 23953333373114771673188958012114092773131100316071109 191
54 953333373114771673188958012114092773131100316071109281 197
55 3333731147716731889580121140927731311003160711092811381 193
56 33731147716731889580121140927731311003160711092811381613 197
57 311477167318895801211409277313110031607110928113816133361 197
58 4771673188958012114092773131100316071109281138161333611103 197
59 71673188958012114092773131100316071109281138161333611103263 197
60 731889580121140927731311003160711092811381613336111032631283 197
61 8895801211409277313110031607110928113816133361110326312834153 199
62 58012114092773131100316071109281138161333611103263128341531697 197
63 121140927731311003160711092811381613336111032631283415316971039 197
64 1140927731311003160711092811381613336111032631283415316971039179 211
65 40927731311003160711092811381613336111032631283415316971039179347 223
66 277313110031607110928113816133361110326312834153169710391793472971 229
67 7313110031607110928113816133361110326312834153169710391793472971151 227
68 13110031607110928113816133361110326312834153169710391793472971151349 233
69 100316071109281138161333611103263128341531697103917934729711513492351 239
70 3160711092811381613336111032631283415316971039179347297115134923513943 257
71 60711092811381613336111032631283415316971039179347297115134923513943541 263
72 110928113816133361110326312834153169710391793472971151349235139435411459 269
73 9281138161333611103263128341531697103917934729711513492351394354114592617 283
74 81138161333611103263128341531697103917934729711513492351394354114592617137 283
75 138161333611103263128341531697103917934729711513492351394354114592617137937 293
...
I'm indebted to Éric Angelini for the seed of the idea.
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