%N a(1)=11. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.
%C If a(n-1) has a single digit then a(n) is simply the smallest missing prime.
%C Leading zeros in s are ignored.
%C The b-file suggests that there are infinitely many primes that do not appear in the sequence. However, there is no proof at present that any particular prime (23, say) never appears.
%C _Alois P. Heinz_ points out that this sequence and A262283 eventually merge (see the latter entry for details). - _N. J. A. Sloane_, Sep 19 2015
%C A variant without the prime number condition: A262356. - _Reinhard Zumkeller_, Sep 19 2015
%H Alois P. Heinz, <a href="/A262282/b262282.txt">Table of n, a(n) for n = 1..705</a>
%e a(1)=11, so s=1, a(2) is smallest missing prime that starts with 1, so a(2)=13. Then s=3, so a(3)=3. Then s is the empty string, so a(4)=2, and so on.
%o import Data.List (isPrefixOf, delete)
%o a262282 n = a262282_list !! (n-1)
%o a262282_list = 11 : f "1" (map show (delete 11 a000040_list)) where
%o f xs pss = (read ys :: Integer) :
%o f (dropWhile (== '0') ys') (delete ys pss)
%o where ys@(_:ys') = head $ filter (isPrefixOf xs) pss
%o -- _Reinhard Zumkeller_, Sep 19 2015
%Y Suggested by A089755. Cf. A262283.
%Y Cf. A262356.
%A _Franklin T. Adams-Watters_ and _N. J. A. Sloane_, Sep 18 2015
%E More terms from _Alois P. Heinz_, Sep 18 2015