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.

12, 13, 15, 17, 42, 43, 45, 47, 62, 63, 65, 67, 82, 83, 85, 87, 92, 93, 95, 97, 123, 143, 147, 153, 167, 183, 423, 443, 447, 453, 467, 483, 497, 623, 637, 643, 647, 653, 667, 683, 697, 813, 817, 823, 843, 847, 853, 867, 873, 883, 913, 917, 923, 937, 943, 947, 953, 967, 983, 997

Offset: 1

Author

Tamas Sandor Nagy, Jun 27 2025

Keywords

• nonn
• base
• fini
• full

Comments

As no leading zeros are allowed, all terms are zeroless.

From Michael S. Branicky, Jun 27 2025: (Start)

Finite since each term with first digit removed must be a member of A024785, which is finite.

Last term here is a(7407) = 6357686312646216567629137. (End)

Examples

			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.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..7407

Crossrefs

Cf. A125664, A125524, A024785.

Programs

Mathematica

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 *)

Python

from sympy import isprime
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)))
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, Jun 27 2025

Python

# uses import and function ok above
from itertools import count, islice, product
def agen():  # generator of terms
    tp = list("23579")  # set of left-truncatable primes
    for d in count(2):
        tpnew = []
        for f in "123456789":
            for e in tp:
                if isprime(int(s:=f+e)):
                    tpnew.append(s)
                if ok(t:=int(f+e)):
                    yield t
        tp = tpnew
        if len(tp) == 0:
            return
afull = list(agen())
print(afull[:60]) # Michael S. Branicky, Jun 27 2025