A371862 Positive integers that can be written as the product of two or more other integers, none of which uses any of the digits in the number itself.

4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 24, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 49, 52, 54, 56, 57, 58, 60, 63, 64, 66, 68, 69, 70, 72, 76, 78, 80, 81, 84, 86, 87, 88, 90, 96, 98, 99, 100, 102, 104, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120

Offset: 1

Author

Bernardo Recamán and Freddy Barrera, Apr 09 2024

Keywords

• nonn
• base

Comments

Infinite since 10^k = 2^k * 5^k and 10^k - 1 = 3^2 * (10^k - 1)/9 are terms for k > 0 and are the smallest (k+1)-digit and largest k-digit terms, resp. All repdigits consisting of 4's, 6's, 8's, or 9's are also terms. - Michael S. Branicky, Apr 09 2024

No number ending in 5 is a term. - Jon E. Schoenfield, Apr 09 2024

Terms are composite. If 9 consecutive positive integers are terms then they are between two consecutive primes at least 14 apart. - David A. Corneth, Apr 10 2024

All products of x (repdigits consisting of 3's or 6's) and 10x + 7 are terms. - Ivan N. Ianakiev, Apr 11 2024

Examples

			60 is a term because it can be expressed as 4 * 15, avoiding its own digits 6 and 0. 50 isn't because there is no way of expressing 50 avoiding both 5 and 0.
112 is a term since 112 = 4*4*7 and is the first term requiring a product with three factors.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Bernardo Recamán Santos, Melissa's Numbers, Puzzling Stack Exchange, Mar 13 2024.

Crossrefs

Cf. A001055, A002808, A162247.

Programs

Python

from sympy import divisors, isprime
from functools import cache
@cache
def ok(n, avoid=tuple()):
    if avoid == tuple(): avoid = set(str(n))
    else: avoid = set(avoid)
    if n%10 == 5 or len(avoid) == 10 or isprime(n): return False
    for d in divisors(n)[1:-1]:
        if set(str(d)) & avoid == set():
            if set(str(n//d)) & avoid == set(): return True
            if ok(n//d, tuple(sorted(avoid))): return True
    return False
print([k for k in range(200) if ok(k)]) # Michael S. Branicky, Apr 10 2024