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A262283 a(1)=2. 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.

%I A262283 #33 Sep 20 2015 15:14:47
%S A262283 2,3,5,7,11,13,31,17,71,19,97,73,37,79,907,701,101,103,307,709,911,
%T A262283 113,131,311,1103,1031,313,137,373,733,331,317,173,739,397,971,719,
%U A262283 191,919,193,937,379,797,977,773,7307,3079,7901,9011,1109,109,929,29,941,41
%N A262283 a(1)=2. 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 A262283 If a(n-1) has a single digit then a(n) is simply the smallest missing prime.
%C A262283 Leading zeros in s are ignored.
%C A262283 The sequence is infinite, since there infinitely many primes that start with s (see the comments in A080165).
%C A262283 The data in the b-file suggests that there are infinitely many primes that do not appear. Hoever, at present that is no proof that even one prime (23, say) never appears. - _N. J. A. Sloane_, Sep 20 2015
%C A262283 _Alois P. Heinz_ points out that a(n) = A262282(n+29) starting at the 103rd term. - _N. J. A. Sloane_, Sep 19 2015
%H A262283 Alois P. Heinz, <a href="/A262283/b262283.txt">Table of n, a(n) for n = 1..676</a>
%e A262283 a(1)=2, so s is the empty string, so a(2) is the smallest missing prime, 3. After a(6)=13, s=3, so a(7) is the smallest missing prime that starts with 3, which is 31.
%o A262283 (Haskell)
%o A262283 import Data.List (isPrefixOf, delete)
%o A262283 a262283 n = a262283_list !! (n-1)
%o A262283 a262283_list = 2 : f "" (map show $ tail a000040_list) where
%o A262283    f xs pss = (read ys :: Integer) :
%o A262283               f (dropWhile (== '0') ys') (delete ys pss)
%o A262283               where ys@(_:ys') = head $ filter (isPrefixOf xs) pss
%o A262283 -- _Reinhard Zumkeller_, Sep 19 2015
%Y A262283 Cf. A080165, A089755, A262282, A262350.
%K A262283 nonn,base
%O A262283 1,1
%A A262283 _N. J. A. Sloane_, Sep 18 2015
%E A262283 More terms from _Alois P. Heinz_, Sep 18 2015