Java generate and track small reusable unique numbers for business use cases

Question

For inventory management we use a key/id called item number

It is a consecutive number starting from 1
It must be unique
The datatype is int
It is not a database key
It should get reused after deletion
We want to keep it small, so customers may enter the number by hand (e.g. the barcode scanner fails or the item label is damaged)

Current situation

Currently we keep track of that number in the database.
On each system (re-)start the highest currently used number is determined and used as a new starting point for every new number we need to generate
if the next number is not free, we just skip it and continue until we find a free one (this check is rather expensive since a query to the db is made; however most of the time the numbers are not in use)
the method responsible for the number generation is serialized for concurrent use

Problem 1: We are not able to reuse any item numbers after they have been deleted

Problem 2: Importing data from other sources

items from foreign systems have their own item number which we need to keep (except if the item number is used in the current system, then a new item number is generated instead)
these item numbers can have a much higher value as those in the current system => this leads to big gaps; after the next system start all new item numbers start from the highest number (!)

As part of system start-up we need to obtain the list of item numbers currently in use; I guess SELECTing just that one column and storing it in a Set should be ok; the required size for that data structure would be approximately 400 KB (a set of 100.000 ids of type int)
This set must then be consulted to find the next minimal number which is not contained in that set; that access needs to be serialized or handled in a concurrent matter (i.e. no two threads are allowed to receive the same number)
I've found a trivial approach on how to obtain the next free number here: Find First missing number in Set
the set and the database must be kept in sync about which numbers are used and which numbers are free; i.e. I could first test the set until I find a free number, and then test in the database; if not free why-so-ever insert in set and repeat ...

Does this approach sound reasonable?
Are you familiar with (standard) libraries which support implementing that kind of number generation?