Let $o$ be an odd positive integer. We say that $o$ is deficient if its sum of positive divisors denoted by $\sigma(o)$ satisfies the inequality $\sigma(o)<2o$. In this case $\sigma(o)=2o-d$ where $d$ is called the deficiency of $o$. If $d $ is odd we say thay $o$ is odd odd deficient.
The list of odd deficient numbers is listed in the OEIS and begins
$1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125$
Solving for the deficiency of some of the given numbers on the list, I found out that their deficiencies $d$ is mostly even.
Since I have no computing tool as of this moment for computing the deficiency of a large numbers on the list. My questions are:
- Among the list, what is the smallest integer whose deficiency is odd. (Answered already:9)
- If there is, can we characterize them? or
- There is no integer in the list with deficiency $d$ odd? (Answer there is)
So question 2 is the one that is needed to be answered.
Thanks in advance.
The odd deficient numbers which have odd deficiency are the odd deficient squares. We have $d=2o-\sigma(o)$, so $d$ will be odd when $\sigma(o)$ is odd. All the divisors of $o$ are odd, so $\sigma(o)$ will be odd when $o$ has an odd number of divisors. But given a divisor $k, \frac ok$ is also a divisor so they come in pairs unless $o$ is a square.