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.

2, 23, 223, 2203, 22003, 220013, 2200103, 22000103, 223000103, 2230001003, 22300010023, 223000100023, 2230001000203, 22301001000203, 223010001000203, 2230010001000203, 22300010001000203, 222300010001000203, 2223000100010001203, 22203000100010001203, 222030001000010001203, 2220300010200010001203

Offset: 1

Author

Bartlomiej Pawlik, Sep 28 2022

Keywords

• nonn
• base

Comments

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.

Sequence construction very similar to A356557 (the difference arises from the order of the conditions).

Length of a(n) is n.

Is the sequence infinite?

Sequence terminates if and only if it contains a term of A125001.

Examples

			a(2) = 23 because the numbers 20, 21, 12, 22 obtained from a(1) = 2 are composite and 23 is a prime.
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.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A356557, A125001, A332603.

Programs

Maple

f:= proc(n) local x, y,z, j;
      for x from 0 to 9 do
        for j from 0 to length(n)-`if`(x=0,1,0) do
           y:= n mod 10^j;
           z:= y + x*10^j + 10*(n-y);
           if isprime(z) then return z fi;
      od od;
      FAIL
end proc:
R:= 2: x:= 2:
for count from 2 to 30 while x <> FAIL do
    x:= f(x); R:= R, x;
od:
R; # Robert Israel, Sep 29 2022

Mathematica

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 *)

Python

from sympy import isprime
from itertools import islice
def anext(an):
    s = str(an)
    for c in "0123456789":
        for k in range(len(s)+1):
            w = s + c if k == 0 else s[:-k] + c + s[-k:]
            if w[0] != "0" and isprime(int(w)): return int(w)
def agen(an=2):
    while an != None: yield an; an = anext(an)
print(list(islice(agen(), 22))) # Michael S. Branicky, Sep 29 2022

Extensions

More terms from Robert Israel, Sep 29 2022