"LSV (the Least Significant Value) is the value of some parameter or function of parameters that would produce a certain p-value alpha. Said another way, you want to know how small an effect would be declared significant at some p-value alpha. The LSV provides a measuring stick for significance on the scale of the parameter, rather than on a probability scale. It shows how sensitive the design and data are. LSN (the Least Significant Number) is the total number of observations that would produce a specified p-value alpha given that the data has the same form. The LSN is defined as the number of observations needed to reduce the variance of the estimates enough to achieve a significant result with the given values of alpha, sigma, and delta (the significance level, the standard deviation of the error, and the effect size). If you need more data to achieve significance, the LSN helps tell you how many more. The LSN is the total number of observations that yields approximately 50% power. Power is the probability of getting significance (p-value < alpha) when a real difference exists between groups. It is a function of the sample size, the effect size, the standard deviation of the error, and the significance level. The power tells you how likely your experiment is to detect a difference (effect size), at a given alpha level. Note: When there are only two groups in a one-way layout, the LSV computed by the power facility is the same as the least significant difference (LSD) shown in the multiple-comparison tables."
You might want to look at the sample size and power analysis platform (In JMP14: DOE > Design Diagnostics > Sample Size and Power).
This result agrees with the power analysis in your oneway analysis that to have a 50% chance of seeing a significant difference (p<0.05) when the true difference is 0.02902 or greater and with a std dev of 0.031386 you would need 21 samples in total. Just note that the difference to detect is defined differently in the two platforms: difference from overall mean in Oneway and difference between groups in Sample Size and Power.
50% power may not be enough. 80% is often the required power and that would require 39 samples.
This may seem like a lot but you need to consider that your within group std dev is large compared to the size of difference you want to detect.