N as the sum of two composite numbers

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A method to find the number of representations of the number N as the sum of two composite numbers (which are of the form 6k-1 or 6k+1), where one of the summands is divisible by P₁ and the other by P₂.

N gcd (P_1 U P_2) = 1

[[tex]\frac{N}{3* P_{1 } * P_{2}}][/tex]

N gcd (P_1 U P_2) ≠ 1

[[tex]\frac{N}{6* P_{1 } * P_{2}}][/tex]

Example:
N = 31416
P_1=17
P_2=101

[[tex]\frac{31416}{3*17*101}[/tex]]=6

Verified by Python code.