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 n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.
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.
First we generate prime number to get the divisors for a number:
In [1]:
n = int(28123)
import pyprimes
p = list(pyprimes.primes_below(n/2))
Write a function to factorize a number:
In [2]:
from bisect import bisect_left
from collections import Counter
def divisors(m):
counter = Counter()
for d in p[:bisect_left(p, m/2)]:
while not m%d:
counter[d] += 1
m = m/d
return(counter)
function to get the product of the divisors:
In [3]:
from numpy import prod
from itertools import combinations
from itertools import chain
def d(m):
d = divisors(m)
return sum(set(chain.from_iterable( (prod(i) for i in combinations(d.elements(), k)) for k in range(1, sum(d.values())))))+1
generate all products for numbers less than n:
In [5]:
import frogress
a = dict()
for i in frogress.Bar(range(1, n), steps=n):
a[i] = d(i)
acquire the abundent numbers only:
In [6]:
abundent={k for k,v in a.items() if v>k}
In [8]:
max(abundent)
Out[8]:
generate all sums of abundent numbers
In [32]:
from itertools import combinations, chain
b = set( chain( (sum(i) for i in combinations(abundent,2) if sum(i) < n),
(2*i for i in abundent if 2*i < n)))
sum for all numbers that are not sum of abundent:
In [34]:
sum(i for i in range(1,n) if i not in b)
Out[34]: