Relation between Nekrasov functions and Bohr–Sommerfeld periods in the pure $SU(N)$ case

We investigate the duality between the Nekrasov function and the quantized Seiberg–Witten prepotential. We test the hypothesis more thoroughly than has yet been done and do not discuss the motivation and historical context of this duality. We verify the conjecture analytically up to $o(\hbar^6, \ln\Lambda)$ for arbitrary $N$ (giving explicit formulas. Moreover, we present the calculation details that are needed for verification using a computer. This allows verifying the conjecture up to $\hbar^6$ and polynomial degrees of $\Lambda$ for $N=2,3,4$. We consider only the case of the pure $SU(N)$ gauge theory.