A366826 Composite numbers whose proper substrings (of their decimal expansions) are all primes.

4, 6, 8, 9, 22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 237, 537, 737

Offset: 1

Author

Kalle Siukola, Oct 25 2023

Keywords

• base
• fini
• full
• nonn

Comments

There are no terms greater than 999 because the only three-digit prime whose substrings are all primes is 373 (see A085823) and prepending or appending any prime digit to it would create a different three-digit substring.

Examples

			237 is included because it is composite and 2, 3, 7, 23 and 37 are all primes.
4 is included because it is composite and has no proper substrings.
		

Crossrefs

Subsequence of A002808.

Cf. A000040.

Cf. A061371, A062115, A202262.

Cf. A033274, A068669, A085823, A279366.

Programs

Python

from itertools import combinations
from sympy import isprime
for n in range(2, 1000):
    if not isprime(n):
        properSubstrings = set(
            int(str(n)[start:end]) for (start, end)
            in combinations(range(len(str(n)) + 1), 2)
        ) - set((n,))
        if all(isprime(s) for s in properSubstrings):
            print(n, end=', ')