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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.

%I A371862 #51 Apr 13 2024 23:00:21
%S A371862 4,6,8,9,10,12,14,16,18,20,21,24,27,28,30,32,34,36,38,40,42,44,46,48,
%T A371862 49,52,54,56,57,58,60,63,64,66,68,69,70,72,76,78,80,81,84,86,87,88,90,
%U A371862 96,98,99,100,102,104,106,108,110,111,112,114,116,117,118,120
%N 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.
%C A371862 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
%C A371862 No number ending in 5 is a term. - _Jon E. Schoenfield_, Apr 09 2024
%C A371862 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
%C A371862 All products of x (repdigits consisting of 3's or 6's) and 10x + 7 are terms. - _Ivan N. Ianakiev_, Apr 11 2024
%H A371862 Michael S. Branicky, <a href="/A371862/b371862.txt">Table of n, a(n) for n = 1..10000</a>
%H A371862 Bernardo Recamán Santos, <a href="https://puzzling.stackexchange.com/questions/125960/melissas-numbers">Melissa's Numbers</a>, Puzzling Stack Exchange, Mar 13 2024.
%e A371862 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.
%e A371862 112 is a term since 112 = 4*4*7 and is the first term requiring a product with three factors.
%o A371862 (Python)
%o A371862 from sympy import divisors, isprime
%o A371862 from functools import cache
%o A371862 @cache
%o A371862 def ok(n, avoid=tuple()):
%o A371862     if avoid == tuple(): avoid = set(str(n))
%o A371862     else: avoid = set(avoid)
%o A371862     if n%10 == 5 or len(avoid) == 10 or isprime(n): return False
%o A371862     for d in divisors(n)[1:-1]:
%o A371862         if set(str(d)) & avoid == set():
%o A371862             if set(str(n//d)) & avoid == set(): return True
%o A371862             if ok(n//d, tuple(sorted(avoid))): return True
%o A371862     return False
%o A371862 print([k for k in range(200) if ok(k)]) # _Michael S. Branicky_, Apr 10 2024
%Y A371862 Cf. A001055, A002808, A162247.
%K A371862 nonn,base
%O A371862 1,1
%A A371862 _Bernardo Recamán_ and _Freddy Barrera_, Apr 09 2024