Euler 0023
The Problem:
GetA perfect number is a number for which the total sum of aits fileproper divisors is exactly equal to the number.
A number is called deficient if the sum of namesits proper divisors is less than and it is called abundant if eachthis lettersum inexceeds.
As 12 is the namesmallest abundant number, the smallest number that can be written as the sum of two abundant numbers is mapped24. fromBy 1-26mathematical andanalysis, thenit multipliedcan 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 positiongreatest innumber anthat alphabeticalcannot sort.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.
Considerations and Approach:
WeBased readoff inof the filequestion intowe anknow arraythat we only have to go up to 28123.
We are going to generate all abundant numbers up to 28123 - 12, which is the the smallest abundant number.
After we generate all abundant numbers, we can generate all of names.the Afternumbers that itare justsums needsof totwo beabundant sortednumbers. Then we subtract that from 28123 and then propagated through to add the letter sum multiplied by the alphabetical position to the total sum.voila.
The Code:
namesimport math
def generate_primes(upper):
prime_threshold = open("names.txt",upper
"r")
name_listnumber_list = sorted(list(range(2, prime_threshold+1))
prime_list = []
current = 0
total_iteration = 0
while current < len(number_list):
if number_list[current] != -1:
prime = number_list[current]
prime_list += [prime]
increment = current + prime
while increment < len(number_list):
number_list[increment] = -1
increment += prime
total_iteration += 1
#print(number_list)
#print(prime_list)
current += 1
return prime_list
def divisor_finder(number, primes):
upper_limit = number
#upper_limit = 20023110541 #, good to use to double check. Is [2,2,3,167]
current_upper_limit = upper_limit
sqrt_limit = math.sqrt(upper_limit)
prime_factors = []
#we are going to hardcode 2 and 3 for looping sake
i = 0
while primes[i] < sqrt_limit and i < len(primes):
while current_upper_limit % primes[i] == 0:
current_upper_limit = current_upper_limit/primes[i]
prime_factors += [primes[i]]
i += 1
if current_upper_limit != 1:
prime_factors += [current_upper_limit]
#print(prime_factors)
prime_counts = [[prime_factors[0], prime_factors.count(prime_factors[0])]]
#print(prime_counts,prime_counts[-1][0],prime_factors[1])
for i in range(1,len(prime_factors)):
#print(prime_counts[-1][0],prime_factors[i])
if prime_counts[-1][0] != prime_factors[i]:
prime_counts += [[prime_factors[i], prime_factors.count(prime_factors[i])]]
#print(prime_counts)
divisors = [1]
counters = [x[1:-1] for x in names.readline().split(",")])prime_counts]
names_score#print(counters)
while sum(counters) > 0:
new_divisor = 01
for i in range(len(name_list)counters)):
current_name#print(prime_counts[i][0],counters[i], new_divisor)
new_divisor *= name_list[prime_counts[i][0]**counters[i]
current_score = 0
for letter in current_name:
#print(letter, ord(letter) - 64)
current_scoredivisors += ord(letter)[new_divisor]
counters[0] -= 1
for i in range(len(counters)):
if counters[i] == -1:
counters[i] = prime_counts[i][1]
if i != len(counters) - 641:
names_scorecounters[i+1] -= 1
else:
break
divisors.sort()
return divisors[:-1]
upper_limit = 28124
prime_list = generate_primes(upper_limit)
abundant_nums = []
for i in range(12, upper_limit-12):
divisors = divisor_finder(i, prime_list)
if sum(divisors) > i:
#print(i,sum(divisors), divisors)
abundant_nums += current_score*(i+1)[i]
print(name_list[i],abundant_nums)
(i+1)*current_score,abundant_sum_nums names_score)= []
for num1 in abundant_nums:
for num2 in abundant_nums:
number = num1 + num2
if number < upper_limit:
abundant_sum_nums += [number]
else:
break
print(len(name_list)abundant_nums))
remove = set(abundant_sum_nums)
keep = [x for x in range(1,upper_limit)]
for number in remove:
keep.remove(number)
print(len(keep))