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Let $\sigma(n)$ be the sum of the divisors of the positive integer $n$, and denote the deficiency of $n$ by $D(n)=2n-\sigma(n)$.

Here is my question:

Do there exist numbers $m$ which (provably) do not have any pre-images under $m=D(n)=2n-\sigma(n)$?

I am currently unable to come up with a specific example.

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    You might think that $3$ and $-1$ might be simpler than most, but apparently $-1$ is unsolved (it must be an odd square, and there are no known examples below $10^{35}$ - see R. K. Guy, Unsolved Problems in Number Theory, Section B2. https://archive.org/stream/springer_10.1007-978-0-387-26677-0#page/n91/mode/2up ) – 2017-01-09
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    @Henry, what is the status for $3$? – 2017-01-09
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    I don't have access to the article, but this seems extremely relevant to the question: http://projecteuclid.org/euclid.mjms/1449161366. The wording (not definitively) suggests that there is probably no $m$ that is currently known to lie outside of the range of $D$. – 2017-01-09
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    Is it known that there is an $m$ that does not appear? – 2017-01-09
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    @barto, I currently do not have an answer to that inquiry; hence, my question in this MSE post. – 2017-01-09
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    Well there are two questions: "Does $D(n)$ reach all integers?" and the more difficult "Which integers do not appear?". The question suggests that the answer to the first one is known to be false. – 2017-01-09
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    Okay, fair enough. Editing my question in response to your last comment, @barto. – 2017-01-09

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