How many numbers less than 100 have the sum of factors as odd?

How many numbers less than 100 have the sum of factors as odd?

Answer is 16

This question and explanation is taken from careerbless.com The link given derives the answer using some properties of number. But, I am trying to solve it in a different way, using a general formula to find sum of the factors of numbers, possibility that may lead to a simple solution.

I have gone through the post in this website where a formula is explained to find sum of the factors of a number. But, it cannot be applied individually for this question (because we need to do it for 1-99 and it takes lot of time).

Is there any shortcut method to derive the answer for this question, possibility using a formula to understand if the sum of the factors is even or odd? Please comment with different approaches for this problem.

Answer

The sum-of-divisors function $\sigma$ is multiplicative, with $\sigma(p^m) = 1 + p + \ldots + p^m$ for a prime $p$. In particular, $\sigma(2^m)$ is always odd, while for an odd prime $p$, $\sigma(p^m)$ is odd if and only if $m$ is even. So $\sigma(x)$ is odd if and only if
$x$ is of the form $2^k y^2$ where $y$ is odd.

EDIT: The numbers such that $\sigma(n)$ is odd form OEIS sequence A028982. The number of $x \le n$ such that
$\sigma(x)$ is odd is OEIS sequence A071860.