Re: "I think there are many factorization contexts where iterating FactorInteger for consecutive numbers in a range is more time consuming than a single execution of a function (or "built-in symbol") based on the algorithm shown here."

For small numbers, in a range where sieving is effective, one might benefit from a pure sieve-based approach. Beyond several digits this will not be optimal. Beyond 20 or so digits it will not even be tractable.

Yes, it is possible to find a relation between the Sieve of Eratosthenes and the algorithm introduced in this discussion.

Proceeding in similar ways Sieve of Eratosthenes allows finding prime numbers while the algorithm finds the exact exponent of each factor in a number, so generating the sequence of all the positive numbers by neatly varying factors... It counts.

The "reverse factoring counter" counts in a counterintuitive way; it is an unreasonable way for the purpose of "finding the next number", but it would be great for the purpose of "knowing the factors of the next number".

Also, given an arbitrary prime, the algorithm can generate the corresponding column of the main table independently from the data in other columns (although it can do that only running from the beginning). "Interrelation of columns" occurs just to check the "next prime needed" condition; and this is the part where the algorithm has nothing new to say with respect to the Sieve of Eratosthenes.

I thank for the observation. I suppose that the Sieve of Eratosthenes behavior has been proven computationally irreducible. Perhaps it is possible to give an answer to the question of this discussion in a similar way.

In this discussion I've given an ECMA-262 definition of the algorithm that describes the behavior of what I called (to fit in title) the "reverse factoring" counter.

Now I give the URL of an HTML5 page showing that behavior by an embedded JavaScript implementation of that algorithm... http://www.comprendonio.info/I/UnlimitedInside/I/ReverseFactoringCounterBehavior.HTM

Observing the graph it seems impossible to predict the behavior, for example, from row 1001 to row 1200, without computing all the steps from the beginning (row 1).

Of course it is possible to factorize 1001 to find all the data of the row 1001; nevertheless it does not seem possible to operate any version of the reverse factoring counter from there forward (again, without running it from the beginning).

Also a reasoning must be done about the known algorithms to factorize positive numbers... Do they covertly embed the rule I set in my algorithm? (Or vice versa, do my algorithm covertly embed factoring rules?)

For sure, determining every data in the table from row 1001 to row 1200 by factoring numbers cannot be considered a prove of computational reducibility of the behavior shown by the reverse factoring counter.

Anyway the question in the title of this discussion remain open because some "partial predictions" are suggested by the numerical version of the table (in the linked page). I'm going to think about them.