Restricted Factorisations
Problem 636
Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring different base numbers for each power.
For example, can be written as a product of a square and a fourth power in three ways such that the base numbers are different.
That is,
That is,
Though and are both equal, we are concerned only about the base numbers in this problem. Note that permutations are not considered distinct, for example and are considered to be the same.
Similarly, can be written as a product of one natural number, two squares and three cubes in two ways () whereas can be given the same representation in ways.
Let denote the number of ways in which can be written as a product of one natural number, two squares, three cubes and four fourth powers.
You are given that , ,
and .
and .
Find .