-- A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

--

-- A number whose proper divisors are less than the number is called deficient and a number whose proper divisors exceed the number is called abundant.

--

-- As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

--

-- Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.



require "euler"



local last = 28123

primes, prime_table = sieve_primes(last)



function sum(t, b, e)

  local s = 1

  local i

  for i = b, e do

    local p = primes[i]

    local x = t[i]

    local m = 1

    for k = 1, x do

      m = m + math.pow(p, k)

    end

    s = s * m

  end

  return s

end



abundance = {}



function is_abundant(i)

  local x = abundance[i]

  if x==nil then 

    local d, n = prime_divisors(i, primes)

    x = sum(d, 1, n) - i

    abundance[i] = x

  end

  -- print(i, x)

  return x>i

end



function is_perfect(i)

  local x = abundance[i]

  if x==nil then 

    local d, n = prime_divisors(i, primes)

    x = sum(d, 1, n) - i

    abundance[i] = x

  end

  -- print(i, x)

  return x==i

end



function test(n)

  local x, l = prime_divisors(n, primes)

  local k, v

  for k, v in pairs(x) do print(v, primes[k]) end

  k = sum(x, 1, l) - n

  return k

end



-- print(test(284))

assert(test(284)==220)

assert(test(4*9)==91-4*9)

assert(test(6)==1+2+3)

assert(test(12)==1+2+3+4+6)

assert(test(2)==1)

assert(test(4)==3)

assert(test(220)==284)



assert(is_perfect(28))

assert(is_abundant(12))

assert(not is_abundant(28))



local i

local a, abundant = 0, {}

for i=12,last do

  if is_abundant(i) then

    a = a + 1

    abundant[a] = i

  end

end



-- for k, v in ipairs(abundant) do print(v) end



local sum = 0

for i=1,last do

  local k, v

  local found = false

  for k, v in ipairs(abundant) do

    local r = i-v

    if r<v then break end

    if is_abundant(r) then 

      found = true

      break

    end

  end

  if not found then

    sum = sum + i

  end

end

print(sum)