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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).

%I A125001 #23 Feb 26 2024 19:16:52
%S A125001 369293,3823867,5364431,5409259,7904521,8309369,9387527,9510341,
%T A125001 22038829,27195601,28653263,38696543,39091441,39113161,43744697,
%U A125001 45095839,45937109,48296921,48694231,49085093,49106677,50791927
%N 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).
%C A125001 Is the sequence infinite? - _Zak Seidov_, Nov 14 2014
%H A125001 David W. Wilson and Zak Seidov, <a href="/A125001/b125001.txt">Table of n, a(n) for n = 1..3000</a>
%H A125001 Jeremiah T. Southwick, <a href="https://scholarcommons.sc.edu/etd/5879/">Two Inquiries Related to the Digits of Prime Numbers</a>, Ph. D. Dissertation, University of South Carolina (2020).
%e A125001 369293 is a member because all of 1369293, 2369293, 3369293, ..., 3069293, 3169293, ..., 3692930, ..., 3692939 are composite.
%t A125001 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 *)
%o A125001 (Python)
%o A125001 from sympy import isprime
%o A125001 from itertools import islice
%o A125001 def ok(n):
%o A125001     if not isprime(n): return False
%o A125001     s = str(n)
%o A125001     for c in "0123456789":
%o A125001         for k in range(len(s)+1):
%o A125001             w = s + c if k == 0 else s[:-k] + c + s[-k:]
%o A125001             if w[0] != "0" and isprime(int(w)): return False
%o A125001     return True
%o A125001 print([k for k in range(10**7) if ok(k)]) # _Michael S. Branicky_, Sep 29 2022
%Y A125001 Cf. A356557, A357436.
%K A125001 nonn,base
%O A125001 1,1
%A A125001 _David W. Wilson_, Jan 08 2007