Inspired by Mehul Chaturvedi

Chemistry Level 4

\[P(x)=\dfrac{ax^7}{b}+\dfrac{cx^6}{d}+\dfrac{ex^5}{f}+\dfrac{gx^4}{h}+\dfrac{ix^3}{j}+\dfrac{kx^2}{l}+\dfrac{mx}{n}-22\]

Let a polynomial \(P(x)\) be defined such that \(P(n)\) is the number of open-chain structural isomers of \(n\) carbon membered alkanes for \(n\in [1,8]\).

Given that \(a,c,e,g,i,k\) and \(m\) are positive integers such that \( \gcd(a,b), \gcd(c,d) , \gcd(e,f) , \gcd(g,h) ,\gcd(i,j) , \gcd(k,l) , \gcd(m,n) \) are all equal to 1.

Calculate \(a+b+c+d+e+f+g+h+i+j+k+l+m+n-22\).

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