I am thinking if it is possible to define some interesting sequences from the sum of remainders function $$\rho(n)=\sum_{k=1}^n\text{nmod k}$$ (example: $S(4)=\text{4 mod 1 +4 mod 2+4 mod 3+4 mod 4=0+0+1+0=1}$) like as those that were in the literature for superabundant numbers or colosally abundant numbers (this ñast definition from the previous one, introducing an $\epsilon>0$ in the first definition, the definition of the superabundant numbers).
The sum of remainders function satisfies, for $n>1$, $$\sigma(n)=\rho(n-1)-\rho(n)+2n-1,$$ you can find it for example in Spivey, The Humble Sum of Remainders Function (see the proof of Theorem 1), Mathematics MAGAZINE 78(4), 2005, or also for example in the reference to Cross.
Let the arithmetical function defined by $g(n)=\frac{\rho(n-1)-\rho(n)}{n}$, then $n$ is superabundant if $$g(n)-\frac{1}{n}>g(m)-\frac{1}{m}$$ for all $m<n$.
I know that for positive integers one has $N<M$ if and only if $-1/N>-1/M$. The interest of such kind of sequences that were in the literaure is related with unsolved problem in the following ways, if I understand such: 1) inclusion relationships of sets of such sequences, 2) if there are infinitely many terms of such sequences and 3) problems related with the fact is some functions are decreasing functions over the set that define the sequence itself.
Question. What's is your proposal of an interesting sequence (don't need to be related to unsolved problems) deduced/involving from the sum of remainders function and a kind of positive integers with abundancy of divisors (superabundant numbers, colosally abundant numbers...)? Can you explain us your reasoning, that how do you deduce your definition and why is interesting, and do the first experiment to get the first terms of your sequence? Thanks in advance.
I am interesting in this question to learn how explore conjectures and how runs the mathematical reasoning.