A357436 -id:A357436 - cp's OEIS frontend

A125001 Non-insertable primes: primes with property that no matter where you insert (or prepend or append) a digit you get a composite number (except for prepending a zero).

369293, 3823867, 5364431, 5409259, 7904521, 8309369, 9387527, 9510341, 22038829, 27195601, 28653263, 38696543, 39091441, 39113161, 43744697, 45095839, 45937109, 48296921, 48694231, 49085093, 49106677, 50791927

Offset: 1

David W. Wilson, Jan 08 2007

Is the sequence infinite? - Zak Seidov, Nov 14 2014

			369293 is a member because all of 1369293, 2369293, 3369293, ..., 3069293, 3169293, ..., 3692930, ..., 3692939 are composite.

David W. Wilson and Zak Seidov, Table of n, a(n) for n = 1..3000
Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, Ph. D. Dissertation, University of South Carolina (2020).

Cf. A356557, A357436.

nipQ[x_]:=Module[{id=IntegerDigits[x],len},len=Length[id];AllTrue[ Select[ Flatten[Table[FromDigits[Insert[id,n,i]],{i,len+1},{n,0,9}],1],#!=x&], CompositeQ]]; Select[ Prime[Range[3050000]],nipQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 12 2018 *)

from sympy import isprime
from itertools import islice
def ok(n):
    if not isprime(n): return False
    s = str(n)
    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 False
    return True
print([k for k in range(10**7) if ok(k)]) # Michael S. Branicky, Sep 29 2022

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.

Original entry on oeis.org

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

Offset: 1

Bartlomiej Pawlik, Aug 12 2022

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.

			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.

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

Cf. A125001, A332603, A357436.

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 *)

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

cp's OEIS Frontend

A125001 Non-insertable primes: primes with property that no matter where you insert (or prepend or append) a digit you get a composite number (except for prepending a zero).

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