#### /23.lua

Lua | 101 lines | 78 code | 12 blank | 11 comment | 12 complexity | b321b53fa6ea9eb395085b10eeda590a MD5 | raw file
```  1-- 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.
2--
3-- 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.
4--
5-- 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.
6--
7-- Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
8
9require "euler"
10
11local last = 28123
12primes, prime_table = sieve_primes(last)
13
14function sum(t, b, e)
15  local s = 1
16  local i
17  for i = b, e do
18    local p = primes[i]
19    local x = t[i]
20    local m = 1
21    for k = 1, x do
22      m = m + math.pow(p, k)
23    end
24    s = s * m
25  end
26  return s
27end
28
29abundance = {}
30
31function is_abundant(i)
32  local x = abundance[i]
33  if x==nil then
34    local d, n = prime_divisors(i, primes)
35    x = sum(d, 1, n) - i
36    abundance[i] = x
37  end
38  -- print(i, x)
39  return x>i
40end
41
42function is_perfect(i)
43  local x = abundance[i]
44  if x==nil then
45    local d, n = prime_divisors(i, primes)
46    x = sum(d, 1, n) - i
47    abundance[i] = x
48  end
49  -- print(i, x)
50  return x==i
51end
52
53function test(n)
54  local x, l = prime_divisors(n, primes)
55  local k, v
56  for k, v in pairs(x) do print(v, primes[k]) end
57  k = sum(x, 1, l) - n
58  return k
59end
60
61-- print(test(284))
62assert(test(284)==220)
63assert(test(4*9)==91-4*9)
64assert(test(6)==1+2+3)
65assert(test(12)==1+2+3+4+6)
66assert(test(2)==1)
67assert(test(4)==3)
68assert(test(220)==284)
69
70assert(is_perfect(28))
71assert(is_abundant(12))
72assert(not is_abundant(28))
73
74local i
75local a, abundant = 0, {}
76for i=12,last do
77  if is_abundant(i) then
78    a = a + 1
79    abundant[a] = i
80  end
81end
82
83-- for k, v in ipairs(abundant) do print(v) end
84
85local sum = 0
86for i=1,last do
87  local k, v
88  local found = false
89  for k, v in ipairs(abundant) do
90    local r = i-v
91    if r<v then break end
92    if is_abundant(r) then
93      found = true
94      break
95    end
96  end