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I have a number that is perfect square, for example 36, and I have all prime divisors of this number 2,2,3,3. How many ways do we have to get 36 from divisors if we need to multiply k(from 2 to 1000) elements? (repetition is allowed, order matters) For example, if k = 3: 1,6,6 - one way to get 36. In addition, we have upper limit for each of element of multiplication. For 36 it will be 6 for example. It means that we can use 1,6,6 combination but we can not use 1,9,4 because 9 > 6.

I've tried Stars and Bars method but it does not take into account upper limit. After that I found Extended stars-and-bars problem(where the upper limit of the variable is bounded) but the problem that each element in our multiplication can be combination of different prime numbers(not just only one)

Ivan
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    You are asking essentially the same question for the third time. Previous times you got correct answers, but they were not the answers you were looking for. This all happens because what you really have to do is to formulate your problem VERY thoroughly, scrupulously and meticulously. Without “for examples”, without essential parts going in brackets and “in addition”. I guess, you are writing some kind of a program. So please take an effort to formulate your technical specifications accordingly, if you want to get an answer that fits your needs. – Aig Feb 28 '24 at 14:20

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