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We investigate the duality between the Nekrasov function and the quantizedSeiberg-Witten prepotential, first guessed in [1] and further elaborated in [2]and [3]. We concentrate on providing more thorough checks than the onespresented in [3] and do not discuss the motivation and historical context ofthis duality. The check of the conjecture up to $o (\hbar^6, \ln (\Lambda))$ isdone by hands for arbitrary $N$ (explicit formulas are presented). Moreover,details of the calculation that are essential for the computerization of thecheck are worked out. This allows us to test the conjecture up to $\hbar^6$ andup to higher powers of $\Lambda$ for $N = 2,3,4$. Only the case of pure SU(N)gauge theory is considered.

On relation between Nekrasov functions and BS periods in pure SU(N) case