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A385404 Numbers that can be split into two at any place between their digits such that the resulting numbers are always a nonprime on the left and a prime on the right.

%I A385404 #20 Jun 27 2025 16:26:33
%S A385404 12,13,15,17,42,43,45,47,62,63,65,67,82,83,85,87,92,93,95,97,123,143,
%T A385404 147,153,167,183,423,443,447,453,467,483,497,623,637,643,647,653,667,
%U A385404 683,697,813,817,823,843,847,853,867,873,883,913,917,923,937,943,947,953,967,983,997
%N A385404 Numbers that can be split into two at any place between their digits such that the resulting numbers are always a nonprime on the left and a prime on the right.
%C A385404 As no leading zeros are allowed, all terms are zeroless.
%C A385404 From _Michael S. Branicky_, Jun 27 2025: (Start)
%C A385404 Finite since each term with first digit removed must be a member of A024785, which is finite.
%C A385404 Last term here is a(7407) = 6357686312646216567629137. (End)
%H A385404 Michael S. Branicky, <a href="/A385404/b385404.txt">Table of n, a(n) for n = 1..7407</a>
%e A385404 637 is a term because when it is split in two in all possible ways, it first results in 63, a nonprime, and 3, a prime. When split in the second and final possible way, it results in 6, a nonprime, and 37, a prime.
%t A385404 q[n_] := !MemberQ[IntegerDigits[n], 0] && AllTrue[Range[IntegerLength[n]-1], PrimeQ[QuotientRemainder[n, 10^#]] == {False, True} &]; Select[Range[10, 1000], q] (* _Amiram Eldar_, Jun 27 2025 *)
%o A385404 (Python)
%o A385404 from sympy import isprime
%o A385404 def ok(n): return '0' not in (s:=str(n)) and len(s) > 1 and all(not isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s)))
%o A385404 print([k for k in range(1000) if ok(k)]) # _Michael S. Branicky_, Jun 27 2025
%o A385404 (Python) # uses import and function ok above
%o A385404 from itertools import count, islice, product
%o A385404 def agen():  # generator of terms
%o A385404     tp = list("23579")  # set of left-truncatable primes
%o A385404     for d in count(2):
%o A385404         tpnew = []
%o A385404         for f in "123456789":
%o A385404             for e in tp:
%o A385404                 if isprime(int(s:=f+e)):
%o A385404                     tpnew.append(s)
%o A385404                 if ok(t:=int(f+e)):
%o A385404                     yield t
%o A385404         tp = tpnew
%o A385404         if len(tp) == 0:
%o A385404             return
%o A385404 afull = list(agen())
%o A385404 print(afull[:60]) # _Michael S. Branicky_, Jun 27 2025
%Y A385404 Cf. A125664, A125524, A024785.
%K A385404 nonn,base,fini,full
%O A385404 1,1
%A A385404 _Tamas Sandor Nagy_, Jun 27 2025