Motivation:
The (principal) value of
$$m\pmod{n}$$
for some positive integers $m> n$, might well be viewed as the value
$$m-\sum_{i=1}^{M_{m,n}}n,\tag{$\Sigma$}$$
for some $M_{m,n}\in \Bbb N$ with $M_{m,n}n\le m$ but $(M_{m,n}+1)n> m$; indeed, we have just subtracted a suitable number $M_{m,n}$, dependent of $n$ and $m$, of $n$s from $m$.
The Questions:
What happens if we replaced our fixed $n$ by a sequence $(a_n)\in \Bbb N^{\Bbb N}$, so that, for instance, for small $m$, where the threshold $M_{m, (a_n)}$ is also low, one substacts only a handful of terms? Has such a thing been studied before and, if so, what is its name? What properties does it have?
Example:
Consider the natural numbers $(n_n)\in\Bbb N^{\Bbb N}$ (of which $0$ is not a member). Then, say,
$$\begin{align} 7\pmod{(n_n)}&=7-\sum_{n=1}^{M_{7, (n_n)}}n \\ &=7-\frac{M_{7, (n_n)}}{2}(M_{7, (n_n)}+1)\\ &=1, \end{align}$$
since $M_{7, (n_n)}=3$ here.
I like how a sequence $(M_{m, (a_n)})_m\in(\Bbb N\cup\{0\})^{\Bbb N}$ is generated.
Please help :)
I think it hasn’t been studied much because it doesn’t have many interesting properties.
I’m going to change the notation a little because it make the analysis simpler - let $s_n$ be the partial sum of the sequence. Then, $m \mod s_n = m - \max_{s_n\leq m} s_n$.
You’d get something like: $$73 \mod 2^n= 9$$ $$9 \mod 2^n= 1$$ $$1 \mod 2^n= 0$$
In the case where $s_n | s_{n+1}$ - this is kind of like using a different base system and decrementing the leading digit (consider the above example in binary). This is particularly neat if you consider $\mod n!$ on numbers while looking at their factorial base.
In general, as the above example notes, this isn’t idempotent (repeated application gives different results), so you won’t have nice properties of various equivalence classes.
However, if there’a nice relations among the $s_n$, I imagine you could also get nice results. For example, modding by the Fibonacci numbers is like stripping the leading term of the Zeckendorf Representation.