# What is the size of the kth Superabundant Number?

Utilizing the data generated by T.D. Noe for the first $$10^6$$ terms of A004394, a plot of $$k$$ against $$log_{10}(S(k))$$ where S(K) is the $$k^{th}$$ Superabundant Number shows a strong linear relationship which suggests the following:

$$S(k)\approx e^{0.242692k}$$

Is this a known result and does this relationship hold past $$k=10^6$$? Can the exact value of the constant be determined?

• Linking to another question where I use this result. – Goldbug Jul 19 at 15:42
• Briggs shows a similar result for the density of SAs but not clear how they are related. See Figure 2. – Goldbug Jul 19 at 15:46