Problem 55345. Easy Sequences 68: Factorial Rectangles
I'll celebrate my comeback to Cody with this one easy problem...
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The rectangle below is special:
Its area is
which equal to
. We call such rectangle a factorial rectangle, which is an integer-sided rectangle with an area equal to a factorial number.
In this problem, we want to know how many are these factorial rectangles.
For a given integer n, we define the function
as the number of factorial rectangles with area
The factorial rectangles with area
are as follows:
, with rotations not allowed. Hence, ![](data:image/png;base64,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)
Write a function that will calculate
, defined as follows:
For
, we are given:
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