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# Arithmetic Number

• Difficulty Level : Hard
• Last Updated : 05 May, 2021

In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. Or in other words, a number N is arithmetic if the number of divisors divides the sum of divisors.
Given a positive integer n. The task is to check whether n is Arithmetic number or not.
Examples:

```Input : n = 6
Output : Yes
Sum of divisor of 6 = 1 + 2 + 3 + 6 = 12.
Number of divisor of 6 = 4.
So, on dividing Sum of divisor by Number of divisor
= 12/4 = 3, which is an integer.

Input : n = 2
Output : No```

Algorithm

1. Find sum of all the factors of a number , say sum.
2. Find count of divisors (say Count).
3. Check if sum is divisible by count.

## C++

 `// CPP program to check if a number is Arithmetic``// number or not``#include ``using` `namespace` `std;` `// Sieve Of Eratosthenes``void` `SieveOfEratosthenes(``int` `n, ``bool` `prime[],``                         ``bool` `primesquare[], ``int` `a[])``{``    ``for` `(``int` `i = 2; i <= n; i++)``        ``prime[i] = ``true``;` `    ``for` `(``int` `i = 0; i <= (n * n + 1); i++)``        ``primesquare[i] = ``false``;` `    ``// 1 is not a prime number``    ``prime[1] = ``false``;` `    ``for` `(``int` `p = 2; p * p <= n; p++) {``        ``// If prime[p] is not changed, then``        ``// it is a prime``        ``if` `(prime[p] == ``true``) {``            ``// Update all multiples of p``            ``for` `(``int` `i = p * 2; i <= n; i += p)``                ``prime[i] = ``false``;``        ``}``    ``}` `    ``int` `j = 0;``    ``for` `(``int` `p = 2; p <= n; p++) {``        ``if` `(prime[p]) {``            ``// Storing primes in an array``            ``a[j] = p;` `            ``// Update value in primesquare[p*p],``            ``// if p is prime.``            ``primesquare[p * p] = ``true``;``            ``j++;``        ``}``    ``}``}` `// Function to count divisors``int` `countDivisors(``int` `n)``{``    ``// If number is 1, then it will have only 1``    ``// as a factor. So, total factors will be 1.``    ``if` `(n == 1)``        ``return` `1;` `    ``bool` `prime[n + 1], primesquare[n * n + 1];` `    ``int` `a[n]; ``// for storing primes upto n` `    ``// Calling SieveOfEratosthenes to store prime``    ``// factors of n and to store square of prime``    ``// factors of n``    ``SieveOfEratosthenes(n, prime, primesquare, a);` `    ``// ans will contain total number of``    ``// distinct divisors``    ``int` `ans = 1;` `    ``// Loop for counting factors of n``    ``for` `(``int` `i = 0;; i++) {` `        ``// a[i] is not less than cube root n``        ``if` `(a[i] * a[i] * a[i] > n)``            ``break``;` `        ``// Calculating power of a[i] in n.``        ``// cnt is power of prime a[i] in n.``        ``int` `cnt = 1;` `        ``// if a[i] is a factor of n``        ``while` `(n % a[i] == 0)``        ``{``            ``n = n / a[i];``            ``cnt = cnt + 1; ``// incrementing power``        ``}` `        ``// Calculating number of divisors``        ``// If n = a^p * b^q then total``        ``// divisors of n``        ``// are (p+1)*(q+1)``        ``ans = ans * cnt;``    ``}` `    ``// if a[i] is greater than cube root of n` `    ``// First case``    ``if` `(prime[n])``        ``ans = ans * 2;` `    ``// Second case``    ``else` `if` `(primesquare[n])``        ``ans = ans * 3;` `    ``// Third casse``    ``else` `if` `(n != 1)``        ``ans = ans * 4;` `    ``return` `ans; ``// Total divisors``}` `// Returns sum of all factors of n.``int` `sumofFactors(``int` `n)``{``    ``// Traversing through all prime factors.``    ``int` `res = 1;``    ``for` `(``int` `i = 2; i <= ``sqrt``(n); i++) {` `        ``int` `count = 0, curr_sum = 1;``        ``int` `curr_term = 1;``        ``while` `(n % i == 0) {``            ``count++;``            ``n = n / i;` `            ``curr_term *= i;``            ``curr_sum += curr_term;``        ``}` `        ``res *= curr_sum;``    ``}` `    ``// This condition is to handle``    ``// the case when n is a prime``    ``// number greater than 2.``    ``if` `(n >= 2)``        ``res *= (1 + n);` `    ``return` `res;``}` `// Check if number is Arithmetic Number``// or not.``bool` `checkArithmetic(``int` `n)``{``    ``int` `count = countDivisors(n);``    ``int` `sum = sumofFactors(n);` `    ``return` `(sum  % count == 0);``}` `// Driven Program``int` `main()``{``    ``int` `n = 6;``    ``(checkArithmetic(n)) ? (cout << ``"Yes"``) :``                           ``(cout << ``"No"``);``    ``return` `0;``}`

## Java

 `// Java program to check if a number is Arithmetic``// number or not``class` `GFG``{``    ` `// Sieve Of Eratosthenes``static` `void` `SieveOfEratosthenes(``int` `n, ``boolean` `prime[],``                        ``boolean` `primesquare[], ``int` `a[])``{``    ``for` `(``int` `i = ``2``; i <= n; i++)``        ``prime[i] = ``true``;` `    ``for` `(``int` `i = ``0``; i <= (n * n ); i++)``        ``primesquare[i] = ``false``;` `    ``// 1 is not a prime number``    ``prime[``1``] = ``false``;` `    ``for` `(``int` `p = ``2``; p * p <= n; p++)``    ``{``        ``// If prime[p] is not changed, then``        ``// it is a prime``        ``if` `(prime[p] == ``true``)``        ``{``            ``// Update all multiples of p``            ``for` `(``int` `i = p * ``2``; i <= n; i += p)``                ``prime[i] = ``false``;``        ``}``    ``}` `    ``int` `j = ``0``;``    ``for` `(``int` `p = ``2``; p <= n; p++)``    ``{``        ``if` `(prime[p])``        ``{``            ``// Storing primes in an array``            ``a[j] = p;` `            ``// Update value in primesquare[p*p],``            ``// if p is prime.``            ``primesquare[p * p] = ``true``;``            ``j++;``        ``}``    ``}``}` `// Function to count divisors``static` `int` `countDivisors(``int` `n)``{``    ``// If number is 1, then it will have only 1``    ``// as a factor. So, total factors will be 1.``    ``if` `(n == ``1``)``        ``return` `1``;` `    ``boolean` `prime[] = ``new` `boolean``[n + ``1``],``            ``primesquare[] = ``new` `boolean``[n * n + ``1``];` `    ``int` `a[] = ``new` `int``[n]; ``// for storing primes upto n` `    ``// Calling SieveOfEratosthenes to store prime``    ``// factors of n and to store square of prime``    ``// factors of n``    ``SieveOfEratosthenes(n, prime, primesquare, a);` `    ``// ans will contain total number of``    ``// distinct divisors``    ``int` `ans = ``1``;` `    ``// Loop for counting factors of n``    ``for` `(``int` `i = ``0``;; i++)``    ``{` `        ``// a[i] is not less than cube root n``        ``if` `(a[i] * a[i] * a[i] > n)``            ``break``;` `        ``// Calculating power of a[i] in n.``        ``// cnt is power of prime a[i] in n.``        ``int` `cnt = ``1``;` `        ``// if a[i] is a factor of n``        ``while` `(n % a[i] == ``0``)``        ``{``            ``n = n / a[i];``            ``cnt = cnt + ``1``; ``// incrementing power``        ``}` `        ``// Calculating number of divisors``        ``// If n = a^p * b^q then total``        ``// divisors of n``        ``// are (p+1)*(q+1)``        ``ans = ans * cnt;``    ``}` `    ``// if a[i] is greater than cube root of n` `    ``// First case``    ``if` `(prime[n])``        ``ans = ans * ``2``;` `    ``// Second case``    ``else` `if` `(primesquare[n])``        ``ans = ans * ``3``;` `    ``// Third casse``    ``else` `if` `(n != ``1``)``        ``ans = ans * ``4``;` `    ``return` `ans; ``// Total divisors``}` `// Returns sum of all factors of n.``static` `int` `sumofFactors(``int` `n)``{``    ``// Traversing through all prime factors.``    ``int` `res = ``1``;``    ``for` `(``int` `i = ``2``; i <= Math.sqrt(n); i++)``    ``{` `        ``int` `count = ``0``, curr_sum = ``1``;``        ``int` `curr_term = ``1``;``        ``while` `(n % i == ``0``)``        ``{``            ``count++;``            ``n = n / i;` `            ``curr_term *= i;``            ``curr_sum += curr_term;``        ``}` `        ``res *= curr_sum;``    ``}` `    ``// This condition is to handle``    ``// the case when n is a prime``    ``// number greater than 2.``    ``if` `(n >= ``2``)``        ``res *= (``1` `+ n);` `    ``return` `res;``}` `// Check if number is Arithmetic Number``// or not.``static` `boolean` `checkArithmetic(``int` `n)``{``    ``int` `count = countDivisors(n);``    ``int` `sum = sumofFactors(n);` `    ``return` `(sum % count == ``0``);``}` `// Driver Program``public` `static` `void` `main(String[] args)``{``    ``int` `n = ``6``;``    ``if``(checkArithmetic(n))``        ``System.out.println(``"Yes"``);``    ``else``        ``System.out.println(``"No"``);``}``}` `// This code has been contributed by 29AjayKumar`

## Python3

 `# Python3 program to check if``# a number is Arithmetic``# number or not``import` `math` `# Sieve Of Eratosthenes``def` `SieveOfEratosthenes(n, prime,primesquare, a):` `    ``for` `i ``in` `range``(``2``,n``+``1``):``        ``prime[i] ``=` `True``;` `    ``for` `i ``in` `range``((n ``*` `n ``+` `1``)``+``1``):``        ``primesquare[i] ``=` `False``;` `    ``# 1 is not a``    ``# prime number``    ``prime[``1``] ``=` `False``;``    ``p ``=` `2``;` `    ``while` `(p ``*` `p <``=` `n):``        ` `        ``# If prime[p] is``        ``# not changed, then``        ``# it is a prime``        ``if` `(prime[p] ``=``=` `True``):``            ``# Update all multiples of p``            ``for` `i ``in` `range``(p ``*` `2``,n``+``1``,p):``                ``prime[i] ``=` `False``;``        ``p``+``=``1``;` `    ``j ``=` `0``;``    ``for` `p ``in` `range``(``2``,n``+``1``):``        ``if` `(prime[p]):``            ` `            ``# Storing primes in an array``            ``a[j] ``=` `p;` `            ``# Update value in``            ``# primesquare[p*p],``            ``# if p is prime.``            ``primesquare[p ``*` `p] ``=` `True``;``            ``j``+``=``1``;` `# Function to count divisors``def` `countDivisors(n):` `    ``# If number is 1, then it``    ``# will have only 1 as a``    ``# factor. So, total factors``    ``# will be 1.``    ``if` `(n ``=``=` `1``):``        ``return` `1``;` `    ``prime ``=` `[``False``]``*``(n ``+` `2``);``    ``primesquare ``=` `[``False``]``*``(n ``*``n ``+` `3``);` `    ``# for storing primes upto n``    ``a ``=` `[``0``]``*``n;` `    ``# Calling SieveOfEratosthenes``    ``# to store prime factors of``    ``# n and to store square of``    ``# prime factors of n``    ``SieveOfEratosthenes(n, prime,primesquare, a);` `    ``# ans will contain``    ``# total number of``    ``# distinct divisors``    ``ans ``=` `1``;` `    ``# Loop for counting``    ``# factors of n``    ``for` `i ``in` `range``(``0``,``True``):` `        ``# a[i] is not less``        ``# than cube root n``        ``if` `(a[i] ``*` `a[i] ``*` `a[i] > n):``            ``break``;` `        ``# Calculating power of``        ``# a[i] in n. cnt is power``        ``# of prime a[i] in n.``        ``cnt ``=` `1``;` `        ``# if a[i] is a factor of n``        ``while` `(n ``%` `a[i] ``=``=` `0``):``            ``n ``/``/``=` `a[i];``            ` `            ``# incrementing power``            ``cnt ``=` `cnt ``+` `1``;` `        ``# Calculating number of``        ``# divisors. If n = a^p * b^q``        ``# then total divisors``        ``# of n are (p+1)*(q+1)``        ``ans ``=` `ans ``*` `cnt;` `    ``# if a[i] is greater``    ``# than cube root of n` `    ``# First case``    ``if` `(prime[n]):``        ``ans ``=` `ans ``*` `2``;` `    ``# Second case``    ``elif` `(primesquare[n]):``        ``ans ``=` `ans ``*` `3``;` `    ``# Third casse``    ``elif` `(n !``=` `1``):``        ``ans ``=` `ans ``*` `4``;` `    ``return` `ans; ``# Total divisors` `# Returns sum of``# all factors of n.``def` `sumofFactors(n):` `    ``# Traversing through``    ``# all prime factors.``    ``res ``=` `1``;``    ``for` `i ``in` `range``(``2``,``int``(math.sqrt(n))``+``1``):``        ``count ``=` `0``;``        ``curr_sum ``=` `1``;``        ``curr_term ``=` `1``;``        ``while` `(n ``%` `i ``=``=` `0``):``            ``count``+``=``1``;``            ``n ``/``/``=` `i;` `            ``curr_term ``*``=` `i;``            ``curr_sum ``+``=` `curr_term;` `        ``res ``*``=` `curr_sum;` `    ``# This condition is to handle``    ``# the case when n is a prime``    ``# number greater than 2.``    ``if` `(n >``=` `2``):``        ``res ``*``=` `(``1` `+` `n);` `    ``return` `res;` `# Check if number is``# Arithmetic Number or not.``def` `checkArithmetic(n):` `    ``count ``=` `countDivisors(n);``    ``sum` `=` `sumofFactors(n);` `    ``return` `(``sum` `%` `count ``=``=` `0``);` `# Driver code``n ``=` `6``;``if``(checkArithmetic(n)):``    ``print``(``"Yes"``);``else``:``    ``print``(``"No"``);` `# This code is contributed``# by mits`

## C#

 `// C# program to check if a number``// is arithmetic number or not``using` `System;` `class` `GFG``{``    ` `// Sieve Of Eratosthenes``static` `void` `SieveOfEratosthenes(``int` `n, ``bool` `[]prime,``                        ``bool` `[]primesquare, ``int` `[]a)``{``    ``for` `(``int` `i = 2; i <= n; i++)``        ``prime[i] = ``true``;` `    ``for` `(``int` `i = 0; i <= (n * n ); i++)``        ``primesquare[i] = ``false``;` `    ``// 1 is not a prime number``    ``prime[1] = ``false``;` `    ``for` `(``int` `p = 2; p * p <= n; p++)``    ``{``        ``// If prime[p] is not changed, then``        ``// it is a prime``        ``if` `(prime[p] == ``true``)``        ``{``            ``// Update all multiples of p``            ``for` `(``int` `i = p * 2; i <= n; i += p)``                ``prime[i] = ``false``;``        ``}``    ``}` `    ``int` `j = 0;``    ``for` `(``int` `p = 2; p <= n; p++)``    ``{``        ``if` `(prime[p])``        ``{``            ``// Storing primes in an array``            ``a[j] = p;` `            ``// Update value in primesquare[p*p],``            ``// if p is prime.``            ``primesquare[p * p] = ``true``;``            ``j++;``        ``}``    ``}``}` `// Function to count divisors``static` `int` `countDivisors(``int` `n)``{``    ``// If number is 1, then it will have only 1``    ``// as a factor. So, total factors will be 1.``    ``if` `(n == 1)``        ``return` `1;` `    ``bool` `[]prime = ``new` `bool``[n + 1];``    ``bool` `[]primesquare = ``new` `bool``[n * n + 1];` `    ``int` `[]a = ``new` `int``[n]; ``// for storing primes upto n` `    ``// Calling SieveOfEratosthenes to store prime``    ``// factors of n and to store square of prime``    ``// factors of n``    ``SieveOfEratosthenes(n, prime, primesquare, a);` `    ``// ans will contain total number of``    ``// distinct divisors``    ``int` `ans = 1;` `    ``// Loop for counting factors of n``    ``for` `(``int` `i = 0;; i++)``    ``{` `        ``// a[i] is not less than cube root n``        ``if` `(a[i] * a[i] * a[i] > n)``            ``break``;` `        ``// Calculating power of a[i] in n.``        ``// cnt is power of prime a[i] in n.``        ``int` `cnt = 1;` `        ``// if a[i] is a factor of n``        ``while` `(n % a[i] == 0)``        ``{``            ``n = n / a[i];``            ``cnt = cnt + 1; ``// incrementing power``        ``}` `        ``// Calculating number of divisors``        ``// If n = a^p * b^q then total``        ``// divisors of n``        ``// are (p+1)*(q+1)``        ``ans = ans * cnt;``    ``}` `    ``// if a[i] is greater than cube root of n` `    ``// First case``    ``if` `(prime[n])``        ``ans = ans * 2;` `    ``// Second case``    ``else` `if` `(primesquare[n])``        ``ans = ans * 3;` `    ``// Third casse``    ``else` `if` `(n != 1)``        ``ans = ans * 4;` `    ``return` `ans; ``// Total divisors``}` `// Returns sum of all factors of n.``static` `int` `sumofFactors(``int` `n)``{``    ``// Traversing through all prime factors.``    ``int` `res = 1;``    ``for` `(``int` `i = 2; i <= Math.Sqrt(n); i++)``    ``{` `        ``int` `count = 0, curr_sum = 1;``        ``int` `curr_term = 1;``        ``while` `(n % i == 0)``        ``{``            ``count++;``            ``n = n / i;` `            ``curr_term *= i;``            ``curr_sum += curr_term;``        ``}` `        ``res *= curr_sum;``    ``}` `    ``// This condition is to handle``    ``// the case when n is a prime``    ``// number greater than 2.``    ``if` `(n >= 2)``        ``res *= (1 + n);` `    ``return` `res;``}` `// Check if number is Arithmetic Number``// or not.``static` `bool` `checkArithmetic(``int` `n)``{``    ``int` `count = countDivisors(n);``    ``int` `sum = sumofFactors(n);` `    ``return` `(sum % count == 0);``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `n = 6;``    ``if``(checkArithmetic(n))``        ``Console.WriteLine(``"Yes"``);``    ``else``        ``Console.WriteLine(``"No"``);``}``}` `// This code contributed by Rajput-Ji`

## PHP

 ` ``\$n``)``            ``break``;` `        ``// Calculating power of``        ``// a[i] in n. cnt is power``        ``// of prime a[i] in n.``        ``\$cnt` `= 1;` `        ``// if a[i] is a factor of n``        ``while` `(``\$n` `% ``\$a``[``\$i``] == 0)``        ``{``            ``\$n` `= (int)(``\$n` `/ ``\$a``[``\$i``]);``            ` `            ``// incrementing power``            ``\$cnt` `= ``\$cnt` `+ 1;``        ``}` `        ``// Calculating number of``        ``// divisors. If n = a^p * b^q``        ``// then total divisors``        ``// of n are (p+1)*(q+1)``        ``\$ans` `= ``\$ans` `* ``\$cnt``;``    ``}` `    ``// if a[i] is greater``    ``// than cube root of n` `    ``// First case``    ``if` `(``\$prime``[``\$n``])``        ``\$ans` `= ``\$ans` `* 2;` `    ``// Second case``    ``else` `if` `(``\$primesquare``[``\$n``])``        ``\$ans` `= ``\$ans` `* 3;` `    ``// Third casse``    ``else` `if` `(``\$n` `!= 1)``        ``\$ans` `= ``\$ans` `* 4;` `    ``return` `\$ans``; ``// Total divisors``}` `// Returns sum of``// all factors of n.``function` `sumofFactors(``\$n``)``{``    ``// Traversing through``    ``// all prime factors.``    ``\$res` `= 1;``    ``for` `(``\$i` `= 2;``         ``\$i` `<= sqrt(``\$n``); ``\$i``++)``    ``{``        ``\$count` `= 0;``        ``\$curr_sum` `= 1;``        ``\$curr_term` `= 1;``        ``while` `(``\$n` `% ``\$i` `== 0)``        ``{``            ``\$count``++;``            ``\$n` `= (int)(``\$n` `/ ``\$i``);` `            ``\$curr_term` `*= ``\$i``;``            ``\$curr_sum` `+= ``\$curr_term``;``        ``}` `        ``\$res` `*= ``\$curr_sum``;``    ``}` `    ``// This condition is to handle``    ``// the case when n is a prime``    ``// number greater than 2.``    ``if` `(``\$n` `>= 2)``        ``\$res` `*= (1 + ``\$n``);` `    ``return` `\$res``;``}` `// Check if number is``// Arithmetic Number or not.``function` `checkArithmetic(``\$n``)``{``    ``\$count` `= countDivisors(``\$n``);``    ``\$sum` `= sumofFactors(``\$n``);` `    ``return` `(``\$sum` `% ``\$count` `== 0);``}` `// Driver code``\$n` `= 6;``echo` `(checkArithmetic(``\$n``)) ?``                     ``"Yes"` `: ``"No"``;` `// This code is contributed``// by mits``?>`

## Javascript

 ``
Output:
`Yes`

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