def solve():
longestLength = 0
smallestMember = None
seen = set()
sieve(10**6)
for n in xrange(2, 1000001):
if n in seen:
continue
chainIndices = {n: 0}
i = 0
while n > 1:
n = aliquot(n)
i += 1
if n > 1000000: break
# This ^^ technically doesn't comply with a _prima facie_
# interpretation of the problem, but it makes the program terminate
# in under a minute with the correct answer, so...
if n in chainIndices:
i2 = chainIndices[n]
if i - i2 > longestLength:
chain = [m for m,j in chainIndices.iteritems() if j >= i2]
if all(m <= 1000000 for m in chain):
longestLength = i - i2
smallestMember = min(chain)
break
elif n in seen:
break
seen.add(n)
chainIndices[n] = i
return smallestMember
def solve():
n = 28123
sieve(n+1)
summable = set()
abundant = []
for i in xrange(1, n+1):
if aliquot(i) > i:
abundant.append(i)
summable.update(i+j for j in abundant if i+j <= n)
return n*(n+1) // 2 - sum(summable)
def solve():
sieve(10000)
cache = {}
accum = 0
for i in xrange(2, 10001):
j = aliquot(i)
if j < i and cache.get(j) == i:
accum += i + j
elif j > i:
cache[i] = j
return accum
