The observations below narrow down the search space for a sequence of nine consecutive Melissa numbers quite a bit. The first number in the sequence ends in either $06$, $16$, $26$, $56$, $76$, $86$ or $96$. The rest of the number, which is the same for all numbers in the sequence, consists of at most five distinct digits, being $0$, $1$, $9$ and at most two other even digits. First some terminology:
- For a Melissa number $n$, a Melissa factorization is a factorization $n=d_1\times\cdots\times d_k$, where the $d_i$ share no digits with $n$. We call each $d_i$ a Melissa factor of $n$.
- For a Melissa number of $m$ digits, we call the last two digits its end and the first $m-2$ digits its start.
Observation 1: A Melissa number cannot end in $5$.
Proof. A number $n$ ends in $5$ if and only if it is an odd multiple of $5$. Then in any factorization, one of its factors is an odd multiple of $5$, and hence it ends in $5$. Then it shares a digit with $n$, and so $n$ is not a Melissa number.
This already shows that there is no sequence of ten or more consecutive Melissa numbers.
Observation 2: A sequence of five or more consecutive Melissa numbers cannot have a $5$ in their start, and cannot have a number ending in $50$.
Proof. As a consequence of observation 1, such a sequence contains a Melissa number ending in $0$, which is therefore divisible by $5$. Then one of its Melissa factors is divisible by $5$, and so it ends in $0$ or $5$. It cannot end in $0$, so one of its Melissa factors must end in $5$, and hence the start of the Melissa number cannot contain a $5$, nor can it end in $50$.
Observation 3: A sequence of eight or more consecutive Melissa numbers cannot have any odd digit other than $1$ or $9$ in their start, and cannot have a number ending in $40$ or $70$.
Proof. We already saw in observation $1$ that no such sequence contains a $5$ in their start. Such a sequence of length eight contains Melissa numbers ending in $3$ and $7$. Any Melissa factorization of a number ending in $3$ contains only odd factors, none of which end in $3$ or $5$. Products of factors ending in $1$ or $9$ also end in $1$ or $9$, and hence there must be a Melissa factor ending in $7$. It follows that the Melissa number does not contain any $7$, so none of the Melissa numbers contain a $7$ in their start, and in particular it does not end in $73$, so the sequence does not contain a number ending in $70$.
The same argument shows that any Melissa number ending in $7$ must have a Melissa factor ending in $3$, so none of the Melissa numbers contain a $3$ in their start, and in particular it does not end in $37$, so the sequence does not contain a number ending in $40$.
Observation 4: In a sequence of nine or more consecutive Melissa numbers, their start cannot contain three distinct even nonzero digits.
Proof. Such a sequence contains Melissa numbers ending in $2$, $4$, $6$ and $8$. Any Melissa factorization of a number ending in $2$ must have an even factor not ending in $0$. Then this factor must end in one of $\{4,6,8\}$, and so the start of the Melissa number cannot contain all three digits $\{4,6,8\}$. The same goes for the Melissa numbers ending and $4$, $6$ and $8$, and this concludes the proof.