Smallest Prime with a Twist (A068103) The task at hand is, given a number n, find the smallest prime that starts with AT LEAST n of the number 2 at the beginning of the number. This is a sequence I found on  OEIS (A068103).  

The first 17 numbers in the sequence are given below, if you want more I'll have to actually go implement the sequence, which I don't mind doing.

0  = 2
1  = 2
2  = 223
3  = 2221
4  = 22229
5  = 2222203
6  = 22222223                # Notice how 6 and 7 are the same! 
7  = 22222223                # It must be **AT LEAST** 6, but no more than necessary.
8  = 222222227
9  = 22222222223             # Notice how 9 and 10 are the same!
10 = 22222222223             # It must be **AT LEAST** 9, but no more than necessary.
11 = 2222222222243
12 = 22222222222201
13 = 22222222222229
14 = 222222222222227
15 = 222222222222222043
16 = 222222222222222221


Just thought this would be a cool combination of string manipulation, prime detection and sequences. This is code-golf, lowest byte count will be declared the winner probably at the end of the month.
Java (OpenJDK 8), 164 110 bytes



a-&gt;{int i=0;for(;!(i+"").matches("2{"+a+"}.*")|new String(new char[i]).matches(".?|(..+)\\1+");i++);return i;}


Thanks to @FryAmTheEggman for a bunch of bytes!

Try it online!