A different way to uniquely factorize natural numbers

lynfac.lua

-- Duval's algorithm
function lyndonFac(str)
    local k = 1
    local ret = {}
    while k <= #str do
        local i,j = k, k+1
        while j <= #str and str:sub(i,i) <= str:sub(j,j) do
            if str:sub(i,i) == str:sub(j,j) then
                i = i + 1
            else
                i = k
            end
            j = j + 1
        end
       
        while k < i + 1 do
            local oldk = k
            k = k + j - i
            table.insert(ret, {oldk,k - oldk})
        end
    end
    return ret
end

function bij_to_num(str)
   local ret = 0
   for i=1, #str do
        ret = ret + (tonumber(str:sub(i,i)) + 1) * 2^(#str - i)
    end
    return ret
end

function num_to_bij(num)
   local ret = ""
   local i = 0
   local rem = (num + 1) % 2
   while num > 0 do
      rem = (num + 1) % 2
      ret = "" .. rem .. ret
      num = math.floor(num/2) - rem
   end
   return ret
end

-- Prints the first 50 factorizations
for i=1, 50 do
   local str = num_to_bij(i)
    for _,v in ipairs(lyndonFac(str)) do
        io.write(bij_to_num(str:sub(v[1], v[1]+v[2] - 1)) .. " ")
    end
    print('')
end