Runs of consecutive numbers all of which are rebel numbers A positive integer is said to be a rebel number if it is the product of numbers none of which share any of the original number´s digits. Thus 10 = 2 x 5 is a rebel number, while none of the primes is. Ten consecutive integers cannot be all rebel numbers because one of them will be an odd multiple of 5 (thus terminating in 5) and with at least one factor terminating in 5. Five consecutive rebel numbers are 666 to 670. Do nine consecutive positive integers all of which are rebel exist? Do they exist infinitely often?

Up to $10^5$, the starts of runs of $5$ consecutive rebel numbers are $666, 686, 1908, 2208, 6666, 7886, 11106, 16166, 21998, 66668, 66786, 70886, 77006$, and there are no runs of more than $5$. This is not an answer, but some evidence pointing to the possiblity that runs of more than $5$ are very rare and maybe nonexistent. The sequence $666, 686, 1908, \ldots$ does not seem to be in the OEIS. Maybe it should be.