Problem 14

Problem

The following iterative sequence is defined for the set of positive integers:

• \(n \to n/2\) (\(n\) is even)
• \(n \to 3n + 1\) (\(n\) is odd)

Using the rule above and starting with \(13\), we generate the following sequence: \(13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1.\)

It can be seen that this sequence (starting at \(13\) and finishing at \(1\)) contains \(10\) terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at \(1\).

Which starting number, under one million, produces the longest chain?

NOTE: Once the chain starts the terms are allowed to go above one million.

Julia

function collatz(n::Integer)
    seq = BigInt[]

    s = BigInt(n)

    while s > 1
        push!(seq, s)

        if s % 2 == 0
            s = div(s, 2)
        else
            s = 3*s + 1
        end
    end 

    push!(seq, 1)
    
    return seq
end;

function p14()

    n, max_chain_length = 1, 1

    for i ∈ 1:999_999
        c = collatz(i)
        l = length(c)

        if l > max_chain_length
            n, max_chain_length = i, l
        end
    end

    return "The biggest chain starts in $n and has $max_chain_length"
end;

# p14()

# using BenchmarkTools;
# @benchmark p14()