Python: Calculate Euclid's totient function of a given integer
Python Basic - 1: Exercise-120 with Solution
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ(n) or ϕ(n), and may also be called Euler's phi function.
Write a Python program to calculate Euclid's totient function of a given integer. Use a primitive method to calculate Euclid's totient function.
def gcd(p,q): # Create the gcd of two positive integers. while q != 0: p, q = q, p%q return p def is_coprime(x, y): return gcd(x, y) == 1 def phi_func(x): if x == 1: return 1 else: n = [y for y in range(1,x) if is_coprime(x,y)] return len(n) print(phi_func(10)) print(phi_func(15)) print(phi_func(33))
4 8 20
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Python: Tips of the Day
How to make a flat list out of list of lists?
Given a list of lists l
flat_list = [item for sublist in l for item in sublist]
flat_list =  for sublist in l: for item in sublist: flat_list.append(item)
is faster than the shortcuts posted so far. (l is the list to flatten.) Here is the corresponding function:
flatten = lambda l: [item for sublist in l for item in sublist]
As evidence, you can use the timeit module in the standard library:
$ python -mtimeit -s'l=[[1,2,3],[4,5,6], , [8,9]]*99' '[item for sublist in l for item in sublist]' 10000 loops, best of 3: 143 usec per loop $ python -mtimeit -s'l=[[1,2,3],[4,5,6], , [8,9]]*99' 'sum(l, )' 1000 loops, best of 3: 969 usec per loop $ python -mtimeit -s'l=[[1,2,3],[4,5,6], , [8,9]]*99' 'reduce(lambda x,y: x+y,l)' 1000 loops, best of 3: 1.1 msec per loop
Explanation: the shortcuts based on + (including the implied use in sum) are, of necessity, O(L**2) when there are L sublists -- as the intermediate result list keeps getting longer, at each step a new intermediate result list object gets allocated, and all the items in the previous intermediate result must be copied over (as well as a few new ones added at the end). So, for simplicity and without actual loss of generality, say you have L sublists of I items each: the first I items are copied back and forth L-1 times, the second I items L-2 times, and so on; total number of copies is I times the sum of x for x from 1 to L excluded, i.e., I * (L**2)/2.
The list comprehension just generates one list, once, and copies each item over (from its original place of residence to the result list) also exactly once.
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