Is the integer $0$ a deficient number?

Question

It is well known that the divisors of the integer $0$ are all non

zero-integers numbers ,the sum of those divisors greater than $0$, then is

it a deficient number ?

Thank you for any help

Answer 1

Due to the strangeness of $0$ when it comes to many number-theoretical properties, it is often excluded from the definitions.

In particular, you will find that many or most definitions concerning special numbers - like deficient numbers - will begin with the phrase a positive integer is called deficient if...

Indeed, even if one chooses to include zero in the definition, it is often an uninteresting case to consider altogether which is another reason why it is not included. In this case, what does the fact that zero is or is not a deficient number tell us about the general case? It doesn't tell us anything because the case is special in itself, so we might as well not include it in the definition.

In fact, the unimportance of zero is so paramount that many definitions may not even specify that it applies to strictly positive integers because the inclusion or exclusion of zero makes no difference.

Answer 2

No, it is not. By definition of a deficient number the sum of the divisors must be $ < 2 n$, but $n=0$ in this case and the sum of divisors $\infty$.