A371862 - OEIS

%I #51 Apr 13 2024 23:00:21

%S 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 49,52,54,56,57,58,60,63,64,66,68,69,70,72,76,78,80,81,84,86,87,88,90,

%U 96,98,99,100,102,104,106,108,110,111,112,114,116,117,118,120

%N 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 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 No number ending in 5 is a term. - _Jon E. Schoenfield_, Apr 09 2024

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

%H Michael S. Branicky, <a href="/A371862/b371862.txt">Table of n, a(n) for n = 1..10000</a>

%H Bernardo Recamán Santos, <a href="https://puzzling.stackexchange.com/questions/125960/melissas-numbers">Melissa's Numbers</a>, Puzzling Stack Exchange, Mar 13 2024.

%e 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 112 is a term since 112 = 4*4*7 and is the first term requiring a product with three factors.

%o (Python)

%o from sympy import divisors, isprime

%o from functools import cache

%o @cache

%o def ok(n, avoid=tuple()):

%o     if avoid == tuple(): avoid = set(str(n))

%o     else: avoid = set(avoid)

%o     if n%10 == 5 or len(avoid) == 10 or isprime(n): return False

%o     for d in divisors(n)[1:-1]:

%o         if set(str(d)) & avoid == set():

%o             if set(str(n//d)) & avoid == set(): return True

%o             if ok(n//d, tuple(sorted(avoid))): return True

%o     return False

%o print([k for k in range(200) if ok(k)]) # _Michael S. Branicky_, Apr 10 2024

%Y Cf. A001055, A002808, A162247.

%K nonn,base

%O 1,1

%A _Bernardo Recamán_ and _Freddy Barrera_, Apr 09 2024