A356557 Start with a(1)=2; to get a(n+1) insert in a(n) at the rightmost possible position the smallest possible digit such that the new number is a prime.

2, 23, 233, 2333, 23333, 233323, 2333231, 23332301, 233323001, 2333230019, 23332030019, 233320360019, 2333203600159, 23332036001959, 233320360019569, 2333203600195669, 23332036001956469, 233320360019564269, 2333203600195642469, 23332036001956424629, 233320360019564246269

Offset

1,1

Comments

Extending a number by inserting a prepending "0" is obviously not allowed. Rightmostness of position has precedence over smallness of digit. If no prime extension exists, the sequence terminates.

Sequence inspired by A332603.

Sequence construction very similar to A357436 (the difference arises from the order of the conditions).

Length of a(n) is n.

Is the sequence infinite? The analogous sequences using bases 2, 3, 4, 5 and 7 are finite.

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

Links

Michael S. Branicky, Table of n, a(n) for n = 1..1000 (terms 1..77 from Bartlomiej Pawlik)

Example

a(2) = 23 because the numbers 20, 21, 22 obtained from a(1) = 2 are composite and 23 is a prime.
For n=6, starting from a(5)=23333 and appending a new rightmost digit gives 233330, 233331, ..., 233339 which are not primes. Inserting a digit second-rightmost gives 233303 and 233313 which are also not prime, and 233323 which is prime, so a(6) = 233323.

Mathematica

k = 2; K = {k}; For[n = 1, n <= 20, n++, r = 0; For[p = IntegerLength[k] + 1, p >= 1, p--, If[r == 1, Break[]]; For[d = 0, d <= 9, d++, If[PrimeQ[ m = ToExpression[StringInsert[ToString[k], ToString[d], p]]], If[k != m, k = m, Print["FINITE"]]; AppendTo[K, k]; r = 1; Break[]]]]]; Print[K] (* Samuel Harkness, Sep 29 2022 *)

Prog

(Python)
from sympy import isprime
from itertools import islice
def anext(an):
    s = str(an)
    for k in range(len(s)+1):
        for c in "0123456789":
            w = s + c if k == 0 else 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(), 21))) # Michael S. Branicky, Aug 12 2022

Crossrefs

Cf. A125001, A332603, A357436.

Sequence in context: A037757 A037645 A100893 * A198972 A065122 A127889

Adjacent sequences: A356554 A356555 A356556 * A356558 A356559 A356560

Keyword

nonn,base

Author

Bartlomiej Pawlik, Aug 12 2022

Status

approved