# What is the best way to get all the divisors of a number?

### Question

Here's the very dumb way:

``````def divisorGenerator(n):
for i in xrange(1,n/2+1):
if n%i == 0: yield i
yield n
``````

The result I'd like to get is similar to this one, but I'd like a smarter algorithm (this one it's too much slow and dumb :-)

I can find prime factors and their multiplicity fast enough. I've an generator that generates factor in this way:

(factor1, multiplicity1)
(factor2, multiplicity2)
(factor3, multiplicity3)
and so on...

i.e. the output of

``````for i in factorGenerator(100):
print i
``````

is:

``````(2, 2)
(5, 2)
``````

I don't know how much is this useful for what I want to do (I coded it for other problems), anyway I'd like a smarter way to make

``````for i in divisorGen(100):
print i
``````

output this:

``````1
2
4
5
10
20
25
50
100
``````

UPDATE: Many thanks to Greg Hewgill and his "smart way" :) Calculating all divisors of 100000000 took 0.01s with his way against the 39s that the dumb way took on my machine, very cool :D

UPDATE 2: Stop saying this is a duplicate of this post. Calculating the number of divisor of a given number doesn't need to calculate all the divisors. It's a different problem, if you think it's not then look for "Divisor function" on wikipedia. Read the questions and the answer before posting, if you do not understand what is the topic just don't add not useful and already given answers.

1
89
5/23/2017 10:31:31 AM

Given your factorGenerator function, here is a divisorGen that should work:

``````def divisorGen(n):
factors = list(factorGenerator(n))
nfactors = len(factors)
f = [0] * nfactors
while True:
yield reduce(lambda x, y: x*y, [factors[x][0]**f[x] for x in range(nfactors)], 1)
i = 0
while True:
f[i] += 1
if f[i] <= factors[i][1]:
break
f[i] = 0
i += 1
if i >= nfactors:
return
``````

The overall efficiency of this algorithm will depend entirely on the efficiency of the factorGenerator.

70
5/28/2015 7:01:34 AM

To expand on what Shimi has said, you should only be running your loop from 1 to the square root of n. Then to find the pair, do `n / i`, and this will cover the whole problem space.

As was also noted, this is a NP, or 'difficult' problem. Exhaustive search, the way you are doing it, is about as good as it gets for guaranteed answers. This fact is used by encryption algorithms and the like to help secure them. If someone were to solve this problem, most if not all of our current 'secure' communication would be rendered insecure.

Python code:

``````import math

def divisorGenerator(n):
large_divisors = []
for i in xrange(1, int(math.sqrt(n) + 1)):
if n % i == 0:
yield i
if i*i != n:
large_divisors.append(n / i)
for divisor in reversed(large_divisors):
yield divisor

print list(divisorGenerator(100))
``````

Which should output a list like:

```[1, 2, 4, 5, 10, 20, 25, 50, 100]
```