From 83ec2cda674ead894784e90e2162a5df36ce585b Mon Sep 17 00:00:00 2001 From: formal-acc Date: Tue, 6 Oct 2020 20:56:20 +0530 Subject: [PATCH 1/6] Added solution to Project Euler 69 --- project_euler/problem_69/__init__.py | 0 project_euler/problem_69/solution_69.py | 59 +++++++++++++++++++++++++ 2 files changed, 59 insertions(+) create mode 100644 project_euler/problem_69/__init__.py create mode 100644 project_euler/problem_69/solution_69.py diff --git a/project_euler/problem_69/__init__.py b/project_euler/problem_69/__init__.py new file mode 100644 index 000000000000..e69de29bb2d1 diff --git a/project_euler/problem_69/solution_69.py b/project_euler/problem_69/solution_69.py new file mode 100644 index 000000000000..2050c072a05c --- /dev/null +++ b/project_euler/problem_69/solution_69.py @@ -0,0 +1,59 @@ +#!/usr/bin/env python3 + +""" +Totient maximum +Problem 69 + +Euler's Totient function, φ(n) [sometimes called the phi function], +is used to determine the number of numbers less than n which are relatively prime to n. +For example, as 1, 2, 4, 5, 7, and 8, +are all less than nine and relatively prime to nine, φ(9)=6. + +n Relatively Prime φ(n) n/φ(n) +2 1 1 2 +3 1,2 2 1.5 +4 1,3 2 2 +5 1,2,3,4 4 1.25 +6 1,5 2 3 +7 1,2,3,4,5,6 6 1.1666... +8 1,3,5,7 4 2 +9 1,2,4,5,7,8 6 1.5 +10 1,3,7,9 4 2.5 + +It can be seen that n=6 produces a maximum n/φ(n) for n ≤ 10. + +Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum. +""" + + +def solution() -> int: + """ + Returns solution to problem. + Algorithm: + Find n/φ(n)for all n ≤ 1,000,000 and return the n that attains maximum + + >>> solution() + 999983 + """ + n = 10 ** 6 + + # Precompute phi using product formula (wikilink below) + # https://en.wikipedia.org/wiki/Euler%27s_totient_function#Euler's_product_formula + + phi = list(range(0, n + 1)) + for number in range(2, n + 1): + if phi[number] == number: + phi[number] -= 1 + for multiple in range(number * 2, n + 1, number): + phi[multiple] = (phi[multiple] // number) * (number - 1) + + answer = 6 + for number in range(10, n + 1): + if phi[answer] / answer < phi[number] / number: + answer = number + + return answer + + +if __name__ == "__main__": + print(solution()) From a2109eed0eddf54f2bea3bc7e008e59ffd93da52 Mon Sep 17 00:00:00 2001 From: sarthaka1310 <56290744+sarthaka1310@users.noreply.github.com> Date: Sun, 11 Oct 2020 13:24:23 +0530 Subject: [PATCH 2/6] Accept edits from code review Co-authored-by: Dhruv --- project_euler/problem_69/solution_69.py | 10 ++-------- 1 file changed, 2 insertions(+), 8 deletions(-) diff --git a/project_euler/problem_69/solution_69.py b/project_euler/problem_69/solution_69.py index 2050c072a05c..18fcc104f1f0 100644 --- a/project_euler/problem_69/solution_69.py +++ b/project_euler/problem_69/solution_69.py @@ -1,8 +1,6 @@ -#!/usr/bin/env python3 - """ Totient maximum -Problem 69 +Problem 69: https://projecteuler.net/problem=69 Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. @@ -26,16 +24,12 @@ """ -def solution() -> int: +def solution(n: int = 10 ** 6) -> int: """ Returns solution to problem. Algorithm: Find n/φ(n)for all n ≤ 1,000,000 and return the n that attains maximum - - >>> solution() - 999983 """ - n = 10 ** 6 # Precompute phi using product formula (wikilink below) # https://en.wikipedia.org/wiki/Euler%27s_totient_function#Euler's_product_formula From 56d1a1190cdca694258dbb3f473e71029651fabc Mon Sep 17 00:00:00 2001 From: formal-acc Date: Sun, 11 Oct 2020 18:45:03 +0530 Subject: [PATCH 3/6] Added doctests --- project_euler/problem_69/solution_69.py | 24 ++++++++++++++++++------ 1 file changed, 18 insertions(+), 6 deletions(-) diff --git a/project_euler/problem_69/solution_69.py b/project_euler/problem_69/solution_69.py index 18fcc104f1f0..e193f864cffd 100644 --- a/project_euler/problem_69/solution_69.py +++ b/project_euler/problem_69/solution_69.py @@ -28,11 +28,23 @@ def solution(n: int = 10 ** 6) -> int: """ Returns solution to problem. Algorithm: - Find n/φ(n)for all n ≤ 1,000,000 and return the n that attains maximum + 1. Precompute φ(k) for all natural k, k <= n using product formula (wikilink below) + https://en.wikipedia.org/wiki/Euler%27s_totient_function#Euler's_product_formula + + 2. Find k/φ(k) for all k ≤ n and return the k that attains maximum + + >>> solution(10) + 6 + + >>> solution(100) + 30 + + >>> solution(9973) + 2310 + """ - # Precompute phi using product formula (wikilink below) - # https://en.wikipedia.org/wiki/Euler%27s_totient_function#Euler's_product_formula + assert n > 0, "Enter a natural number" phi = list(range(0, n + 1)) for number in range(2, n + 1): @@ -41,9 +53,9 @@ def solution(n: int = 10 ** 6) -> int: for multiple in range(number * 2, n + 1, number): phi[multiple] = (phi[multiple] // number) * (number - 1) - answer = 6 - for number in range(10, n + 1): - if phi[answer] / answer < phi[number] / number: + answer = 1 + for number in range(1, n + 1): + if (answer / phi[answer]) < (number / phi[number]): answer = number return answer From 2ff4b3e5874fe6e5bdfe9123756538044ae98ea4 Mon Sep 17 00:00:00 2001 From: sarthaka1310 <56290744+sarthaka1310@users.noreply.github.com> Date: Sun, 11 Oct 2020 21:32:06 +0530 Subject: [PATCH 4/6] Renaming and exception handling --- project_euler/problem_69/{solution_69.py => sol1.py} | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) rename project_euler/problem_69/{solution_69.py => sol1.py} (95%) diff --git a/project_euler/problem_69/solution_69.py b/project_euler/problem_69/sol1.py similarity index 95% rename from project_euler/problem_69/solution_69.py rename to project_euler/problem_69/sol1.py index e193f864cffd..6b9cefb00908 100644 --- a/project_euler/problem_69/solution_69.py +++ b/project_euler/problem_69/sol1.py @@ -44,7 +44,8 @@ def solution(n: int = 10 ** 6) -> int: """ - assert n > 0, "Enter a natural number" + if n <= 0: + raise ValueError("Please enter a natural number") phi = list(range(0, n + 1)) for number in range(2, n + 1): From fefce59db725ce89ea0a9955b70e4f52ef013017 Mon Sep 17 00:00:00 2001 From: sarthaka1310 <56290744+sarthaka1310@users.noreply.github.com> Date: Sun, 11 Oct 2020 22:43:39 +0530 Subject: [PATCH 5/6] Apply suggestions from code review Co-authored-by: Dhruv --- project_euler/problem_69/sol1.py | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/project_euler/problem_69/sol1.py b/project_euler/problem_69/sol1.py index 6b9cefb00908..0d5f84fe3cfa 100644 --- a/project_euler/problem_69/sol1.py +++ b/project_euler/problem_69/sol1.py @@ -47,12 +47,11 @@ def solution(n: int = 10 ** 6) -> int: if n <= 0: raise ValueError("Please enter a natural number") - phi = list(range(0, n + 1)) + phi = list(range(n + 1)) for number in range(2, n + 1): - if phi[number] == number: - phi[number] -= 1 - for multiple in range(number * 2, n + 1, number): - phi[multiple] = (phi[multiple] // number) * (number - 1) + phi[number] -= 1 + for multiple in range(number * 2, n + 1, number): + phi[multiple] = (phi[multiple] // number) * (number - 1) answer = 1 for number in range(1, n + 1): From 21650fc49ebff76dbb7f5453b4eeca8911c77b7e Mon Sep 17 00:00:00 2001 From: sarthaka1310 <56290744+sarthaka1310@users.noreply.github.com> Date: Sun, 11 Oct 2020 23:19:40 +0530 Subject: [PATCH 6/6] Edited mistake. --- project_euler/problem_69/sol1.py | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/project_euler/problem_69/sol1.py b/project_euler/problem_69/sol1.py index 0d5f84fe3cfa..d148dd79a777 100644 --- a/project_euler/problem_69/sol1.py +++ b/project_euler/problem_69/sol1.py @@ -45,13 +45,14 @@ def solution(n: int = 10 ** 6) -> int: """ if n <= 0: - raise ValueError("Please enter a natural number") + raise ValueError("Please enter an integer greater than 0") phi = list(range(n + 1)) for number in range(2, n + 1): - phi[number] -= 1 - for multiple in range(number * 2, n + 1, number): - phi[multiple] = (phi[multiple] // number) * (number - 1) + if phi[number] == number: + phi[number] -= 1 + for multiple in range(number * 2, n + 1, number): + phi[multiple] = (phi[multiple] // number) * (number - 1) answer = 1 for number in range(1, n + 1):