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?

Linear Regression of log10(S(k)) against k.

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

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