A397084 Start with a(1) = 2; to get a(n+1) insert in a(n) the smallest possible digit at the leftmost possible position such that the new number is a prime.

2, 23, 223, 2203, 22003, 220013, 2200013, 20200013, 202002013, 2032002013, 20032002013, 200302002013, 2003020020013, 20030200020013, 200302000201013, 2003020002001013, 20030200020001013, 202030200020001013, 2021030200020001013, 20210530200020001013, 202100530200020001013, 2002100530200020001013

Offset

1,1

Comments

This is the left-right counterpart of A357436. It uses the same precedence of smallness of the inserted digit over the insertion position, but considers positions from left to right instead of from right to left.

Together with A356557, A357436, and A397083, this sequence belongs to the family obtained by choosing the precedence between digit smallness and insertion position, and the direction in which insertion positions are considered.

Smallness of the inserted digit has precedence over leftmostness of its position.

Prepending a 0 is not allowed.

If no prime extension exists, the sequence terminates.

a(n) has n decimal digits.

The sequence terminates if and only if it contains a term of A125001.

Is the sequence infinite?

Links

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

Example

a(2) = 23. Starting from a(1) = 2, insertion positions are considered from left to right for each digit. The numbers 20, 21, 22 are composite, while 23 is prime.
a(7) = 2200013. Starting from a(6) = 220013, insertion of the digit 0 is tested from left to right. Prepending 0 is not allowed; 2020013 is composite, while 2200013 is prime.

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, 1+Length[d]}]]; Array[a, 22]

Prog

(Python)
from sympy import isprime
from itertools import islice
def anext(an):
    s = str(an)
    for c in "0123456789":
        for k in range(int(c=="0"), len(s)+1):
            w = s[:k] + c + s[k:]
            if 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, Jun 16 2026

Crossrefs

Cf. A357436, the right-to-left counterpart, in which smallness of the inserted digit also has precedence over the insertion position.

Cf. A397083, which also considers insertion positions from left to right, but gives precedence to leftmostness of position over smallness of the inserted digit.

Cf. A356557, in which rightmostness of position has precedence over smallness of the inserted digit.

Cf. A125001, A332603.

Sequence in context: A068167 A030456 A397083 * A357436 A069837 A376434

Adjacent sequences: A397081 A397082 A397083 * A397088 A397089 A397090

Keyword

nonn,base

Author

Bartlomiej Pawlik, Jun 16 2026

Status

approved