The code below creates a function that produces a plot of both the differences between ascending unique differences between perfect powers, and the logarithm of these differences also

The function

CatalanFunction <- function(n){
  vector <- c()
  vector1 <- c()
    for(i in 1:n){
      for(j in 1:n){
        vector <- c(vector, i^j)
        }
      vector1 <- c(vector1, vector)
    }
  vector1 <- unique(sort(vector1, decreasing = FALSE))
  vector1 <- diff(vector1, lag = 1)
  vector2 <- c(1:length(vector1))
  vector3 <- log(vector1)
  par(mfrow = c(1, 2))
  plot(vector2, vector1, xlab = "nth value", ylab = "nth difference between unique perfect powers")
  plot(vector2, vector3, xlab = "nth value", ylab = "log(nth difference between unique perfect powers)")
}

Example output plot

The following example provides the plot for the first 20 integers

CatalanFunction(n = 20)