search for the best number of k that minimizes the absolute value of the error between the mean of datas and the mean of frequency table

Question

I wrote this function which gives me as output a frequency table divided in K classes, the mean of datas and the the mead of the frequency table.

Data_Frame <- function(x, k) {
  if(k<=1) {
print("insert a number greter then 1") }
  else {
    data_range <- range(x)
    interval_width <- (max(data_range)-min(data_range))/k
    cutting_values <- seq (from = min(data_range), 
                       to = max(data_range), 
                       by= interval_width,)
    lower_bounds <- cutting_values[1:k]
    upper_bounds <- cutting_values[2:(k+1)]
    counts <- numeric(length = k)  
      for (k in seq_along(counts)) {
        counts[k] <- length(
        x[which((x>=cutting_values[k]) & (x<=cutting_values[(k+1)]))])
      }
   DF <- data.frame(low.bounds = lower_bounds,
                 up.bounds = upper_bounds,
                 freq = counts)
   Data_m <- mean(x)
   DF_m <- sum((DF$low.bounds+DF$up.bounds)/2*DF$freq)/
           sum(DF$freq)
   
   result <- list(DF, Data_m = Data_m, DF_m = DF_m)
  
   return(result)
     }
   }
### Code for using functions
set.seed(4321)
x <- rnorm(1000, 10, 2)
k <- 5L
result_function_1 <- Data_Frame(x, k)
print(result_function_1)

I have to write a second function which must search for the best number of k that minimizes the absolute value of the error between the mean of datas (Data_m) and the mean of frequency table (DF_m). Starting from k = 2 to K = max_K I have to return the optimal k. Could someone help me out?

thanks