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A357436 Start with a(1)=2; to get a(n+1) insert in a(n) the smallest possible digit at the rightmost possible position such that the new number is a prime.

%I A357436 #34 Jun 12 2023 12:33:19
%S A357436 2,23,223,2203,22003,220013,2200103,22000103,223000103,2230001003,
%T A357436 22300010023,223000100023,2230001000203,22301001000203,
%U A357436 223010001000203,2230010001000203,22300010001000203,222300010001000203,2223000100010001203,22203000100010001203,222030001000010001203,2220300010200010001203
%N A357436 Start with a(1)=2; to get a(n+1) insert in a(n) the smallest possible digit at the rightmost possible position such that the new number is a prime.
%C A357436 Extending a number by inserting a prepending "0" is obviously not allowed. Smallness of digit has precedence over rightmostness of position. If no prime extension exists, the sequence terminates.
%C A357436 Sequence construction very similar to A356557 (the difference arises from the order of the conditions).
%C A357436 Length of a(n) is n.
%C A357436 Is the sequence infinite?
%C A357436 Sequence terminates if and only if it contains a term of A125001.
%H A357436 Michael S. Branicky, <a href="/A357436/b357436.txt">Table of n, a(n) for n = 1..1000</a>
%e A357436 a(2) = 23 because the numbers 20, 21, 12, 22 obtained from a(1) = 2 are composite and 23 is a prime.
%e A357436 For n=6, starting from a(5)=22003 and appending a digit "0" from right to left gives 220030, 220003, 202003, which are not primes. Inserting a digit "1" from right to left gives 220031 which also is not prime, and 220013 which is prime, so a(6) = 220013.
%p A357436 f:= proc(n) local x, y,z, j;
%p A357436       for x from 0 to 9 do
%p A357436         for j from 0 to length(n)-`if`(x=0,1,0) do
%p A357436            y:= n mod 10^j;
%p A357436            z:= y + x*10^j + 10*(n-y);
%p A357436            if isprime(z) then return z fi;
%p A357436       od od;
%p A357436       FAIL
%p A357436 end proc:
%p A357436 R:= 2: x:= 2:
%p A357436 for count from 2 to 30 while x <> FAIL do
%p A357436     x:= f(x); R:= R, x;
%p A357436 od:
%p A357436 R; # _Robert Israel_, Sep 29 2022
%t A357436 a[1]=2; a[n_] := a[n] = Catch@ Block[{p, d = IntegerDigits[a[n-1]]}, Do[p = FromDigits@ Insert[d, c, -i]; If[p > a[n - 1] && PrimeQ[p], Throw@p], {c, 0, 9}, {i, 1 + Length@ d}]]; Array[a, 22] (* _Giovanni Resta_, Oct 13 2022 *)
%o A357436 (Python)
%o A357436 from sympy import isprime
%o A357436 from itertools import islice
%o A357436 def anext(an):
%o A357436     s = str(an)
%o A357436     for c in "0123456789":
%o A357436         for k in range(len(s)+1):
%o A357436             w = s + c if k == 0 else s[:-k] + c + s[-k:]
%o A357436             if w[0] != "0" and isprime(int(w)): return int(w)
%o A357436 def agen(an=2):
%o A357436     while an != None: yield an; an = anext(an)
%o A357436 print(list(islice(agen(), 22))) # _Michael S. Branicky_, Sep 29 2022
%Y A357436 Cf. A356557, A125001, A332603.
%K A357436 nonn,base
%O A357436 1,1
%A A357436 _Bartlomiej Pawlik_, Sep 28 2022
%E A357436 More terms from _Robert Israel_, Sep 29 2022