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David Cantrell gives a good approximation of $\Gamma^{-1}(n)$ on this page.

I'll copy the result here in case that page ever goes down:

 k = the positive zero of the digamma function, approximately 1.461632 c = $\sqrt{2\pi}/e - \Gamma(k)$, approximately 0.036534 L(x) = $\ln((x+c)/\sqrt{2\pi})$ W(x) = Lambert W function ApproxInvGamma(x) = L(x) / W(L(x) / e) + 1/2 

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David Cantrell gives a good approximation of $\Gamma^{-1}(n)$ on this page.

I'll copy the result here in case that page ever goes down:

 k = the positive zero of the digamma function, approximately 1.461632 c = $\sqrt{2\pi}/e - \Gamma(k)$, approximately 0.036534 L(x) = $\ln((x+c)/\sqrt{2\pi})$ W(x) = Lambert W function ApproxInvGamma(x) = L(x) / W(L(x) / e) + 1/2