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

1,1

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.

Links

Table of n, a(n) for n=1..19.

Example

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.

Prog

(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=', ')

Crossrefs

Subsequence of A002808.

Cf. A000040.

Cf. A061371, A062115, A202262.

Cf. A033274, A068669, A085823, A279366.

Sequence in context: A121719 A267509 A162738 * A161600 A032350 A078337

Adjacent sequences: A366823 A366824 A366825 * A366827 A366828 A366829

Keyword

base,fini,full,nonn

Author

Kalle Siukola, Oct 25 2023

Status

approved