# Python Program for Find sum of odd factors of a number

Given a number n, the task is to find the odd factor sum. Examples:

```Input : n = 30
Output : 24
Odd dividers sum 1 + 3 + 5 + 15 = 24

Input : 18
Output : 13
Odd dividers sum 1 + 3 + 9 = 13```

Let p1, p2, … pk be prime factors of n. Let a1, a2, .. ak be highest powers of p1, p2, .. pk respectively that divide n, i.e., we can write n as n = (p1a1)*(p2a2)* … (pkak).

```Sum of divisors = (1 + p1 + p12 ... p1a1) *
(1 + p2 + p22 ... p2a2) *
.............................................
(1 + pk + pk2 ... pkak) ```

To find sum of odd factors, we simply need to ignore even factors and their powers. For example, consider n = 18. It can be written as 2132 and sum of all factors is (1)*(1 + 2)*(1 + 3 + 32). Sum of odd factors (1)*(1+3+32) = 13. To remove all even factors, we repeatedly divide n while it is divisible by 2. After this step, we only get odd factors. Note that 2 is the only even prime.

## python3

 `# Formula based Python3 program ` `# to find sum of all divisors ` `# of n. ` `import` `math ` `# Returns sum of all factors ` `# of n. ` `def` `sumofoddFactors( n ): ` `    `  `   ``# Traversing through all ` `   ``# prime factors. ` `   ``res ``=` `1` `    `  `   ``# ignore even factors by ` `   ``# of 2 ` `   ``while` `n ``%` `2` `=``=` `0``: ` `       ``n ``=` `n ``/``/` `2` `    `  `   ``for` `i ``in` `range``(``3``, ``int``(math.sqrt(n) ``+` `1``)): ` `        `  `       ``# While i divides n, print ` `       ``# i and divide n ` `       ``count ``=` `0` `       ``curr_sum ``=` `1` `       ``curr_term ``=` `1` `       ``while` `n ``%` `i ``=``=` `0``: ` `           ``count``+``=``1` `            `  `           ``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. ` `   ``if` `n >``=` `2``: ` `       ``res ``*``=` `(``1` `+` `n) ` `    `  `   ``return` `res ` ` `  `# Driver code ` `n ``=` `30` `print``(sumofoddFactors(n)) ` ` `  `# This code is contributed by “Sharad_Bhardwaj”.`

Output:

`24`

Time complexity: O(sqrt(n))

Auxiliary Space: O(1)

Please refer complete article on Find sum of odd factors of a number for more details!

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