Step by Step solution about How to resolve the algorithm Klarner-Rado sequence step by step in the Julia programming language

The provided code in Julia programming language implements the Klarner-Rado function, which generates a sequence of integers with certain properties. Here's a detailed explanation of the code:

1. KlarnerRado Function:

   • The KlarnerRado function takes a positive integer N as its input.
   • It initializes a list K with a single element, [1].
   • It iterates through the numbers from 1 to N:
     • For each i, it sets j to the i-th element of the list K.
     • It computes two numbers: firstadd (2j + 1) and secondadd (3j + 1).
     • It checks if firstadd is less than the last element of K. If it is, it finds the first position pos in K where firstadd is less than the element at that position.
     • It checks if the element at pos is not equal to firstadd. If it's not, it inserts firstadd into K at position pos.
     • It repeats the above steps for secondadd.
   • Finally, it returns the list K.

2. Generating the First 1 Million Klarner-Rado Numbers:

   • The code generates the first 1 million Klarner-Rado numbers by calling KlarnerRado(1_000_000). It stores the result in the variable kr1m.
   • It prints the first 100 Klarner-Rado numbers, formatted with right padding of 4 characters. It prints a newline after every 20 numbers.
   • It also prints the 1000th, 10000th, 100000th, and 1000000th Klarner-Rado numbers, formatted with commas for readability.

3. KlamerRado Function:

   • The KlamerRado function is another implementation of the Klarner-Rado function.
   • It initializes a boolean array kr of size 100 * N. It sets the first element of kr to true.
   • It iterates through the numbers from 1 to 30*N. For each i, if kr[i] is true, it sets kr[2i + 1] and kr[3i + 1] to true.
   • Finally, it returns a list of integers corresponding to the indices of the true elements in the kr array.

4. Generating the First 1 Million Klarner-Rado Numbers Using KlamerRado:

   • The code generates the first 1 million Klarner-Rado numbers using the KlamerRado(1000000) function and stores the result in kr1m.
   • It prints the first 100 Klarner-Rado numbers and the 1000th, 10000th, 100000th, and 1000000th Klarner-Rado numbers, similar to the previous approach.

Overall, the code provides two different implementations of the Klarner-Rado function and showcases its usage to generate and print the first 1 million Klarner-Rado numbers.

using Formatting

function KlarnerRado(N)
    K = [1]
    for i in 1:N
        j = K[i]
        firstadd, secondadd = 2j + 1, 3j + 1
        if firstadd < K[end]
            pos = findlast(<(firstadd), K) + 1
            K[pos] != firstadd && insert!(K, pos, firstadd)
        elseif K[end] != firstadd
            push!(K, firstadd)
        end
        if secondadd < K[end]
            pos = findlast(<(secondadd), K) + 1
            K[pos] != secondadd && insert!(K, pos, secondadd)
        elseif K[end] != secondadd
            push!(K, secondadd)
        end
    end
    return K
end

kr1m = KlarnerRado(1_000_000)

println("First 100 Klarner-Rado numbers:")
foreach(p -> print(rpad(p[2], 4), p[1] % 20 == 0 ? "\n" : ""), enumerate(kr1m[1:100]))
foreach(n -> println("The ", format(n, commas=true), "th Klarner-Rado number is ",
   format(kr1m[n], commas=true)), [1000, 10000, 100000, 1000000])


using Formatting

function KlamerRado(N)
    kr = falses(100 * N)
    kr[1] = true
    for i in 1:30N
        if kr[i]
            kr[2i + 1] = true
            kr[3i + 1] = true
        end
    end
    return [i for i in eachindex(kr) if kr[i]]
end

kr1m = KlamerRado(1000000)

println("First 100 Klarner-Rado numbers:")
foreach(p -> print(rpad(p[2], 4), p[1] % 20 == 0 ? "\n" : ""), enumerate(kr1m[1:100]))
foreach(n -> println("The ", format(n, commas=true), "th Klarner-Rado number is ",
   format(kr1m[n], commas=true)), [1000, 10000, 100000, 1000000])