Sequence A283190: Remainder Enumeration

• Sequence A283190 is defined as the number of distinct remainders when dividing n by all integers from 1 to ⌊n/2⌋, linking modular arithmetic and combinatorial structure.
• It exhibits linear asymptotic growth with an explicit constant c ≈ 0.2296, derived from a prime product-sum formula and a sieving mechanism.
• The sequence features incremental jumps bounded by one and unbounded negative drops, with iterated remainder sets generalizing its modular dynamics.

Sequence A283190 enumerates, for each positive integer $n$, the number of distinct remainders that can be obtained by dividing $n$ by all positive integers $k$ with $1 \leq k \leq \lfloor n / 2 \rfloor$. Formally, for $n \geq 1$, define the set $$S(n) := \{ n \bmod k : 1 \leq k \leq \lfloor n / 2 \rfloor \}$$ and put $s(n) = |S(n)|$. This sequence exhibits a rich interplay between elementary modular arithmetic, analytic number theory, and the combinatorics of iterated division, giving rise to several remarkable properties concerning its growth, local fluctuations, and generalizations.

1. Definition and Basic Properties

Let $n \in \mathbb{N}$ and $k$ range over $1 \leq k \leq \lfloor n / 2 \rfloor$. The sequence is defined by

$$S(n) = \{ n \bmod k : 1 \leq k \leq \lfloor n / 2 \rfloor \},$$

$s(n) = |S(n)|,$

so $s(n)$ counts the number of residue classes realised as $n$ is reduced modulo small divisors $k$.

One immediately observes that $s(n) \leq \left \lfloor n/2 \right \rfloor$, with the inequality being strict except for trivial values. The sequence appears as A283190 in the OEIS, but has only recently received systematic mathematical analysis (Baraskar et al., 28 Aug 2025).

2. Asymptotic Behavior and Main Constant

The central analytic result establishes that $s(n)$ is asymptotically linear in $n$: $$s(n) = c \cdot n + O\left(\frac{n}{\log n \, \log\log n}\right),$$ where $c$ is an explicit constant given by a product-sum formula over the primes: $$c = \sum_p \frac{1}{p(p+1)} \prod_{p' < p} \left(1 - \frac{1}{p'}\right)$$ and numerically $c \approx 0.2296$.

This arises via a sieving mechanism: for each prime $p$, the contribution to $s(n)$ comes with weight $1/(p(p+1))$ further adjusted for the exclusion of remainders previously “captured” at smaller primes. The error term expresses the thinness of those $k$ (relative to $n$) for which $n \bmod k$ yields “exceptional” remainders.

3. Local Behavior: Small Jumps and Unbounded Drops

A notable and somewhat counterintuitive property is that $s(n)$ increases by at most one at each step,

$s(n+1) \leq s(n) + 1,$

for all $n$, while negative jumps are not bounded below: there exist arbitrarily large drops.

Specifically, the authors show that

$\liminf_{n \to \infty} [s(n+1) - s(n)] = -\infty$

and, for odd $n$, the decrease is bounded by $O(\log \log n)$. Most commonly, the difference is $0$ or $-1$, but deeper analysis reveals infinite occurring negative fluctuations.

The change $s(n+1) - s(n)$ is controlled through “non-transferred” remainders, $T(n, n+1)$, defined by

$T(n, n+1) = \{ r \in S(n) : r+1 \notin S(n+1) \},$

where $r+1$ must be the largest proper divisor of $n - r$. For even $n$, for instance,

$$T(n, n+1) = \begin{cases} \left\{ \frac{n-2}{3} \right\} & \text{if } n \equiv 2 \pmod{3} \ \emptyset & \text{otherwise} \end{cases},$$

implying $s(n+1) = s(n)$ or $s(n+1) = s(n) + 1$ in the even case. The possibility of larger negative jumps is associated with composite structural features of $n$.

4. Iterated Remainder Sets and Connections to Pierce Expansions

The sequence admits generalization via “iterated remainder sets,” capturing the dynamics of repeatedly taking modulo operations: $$S_0(n) := \{1, 2, ..., \lfloor n/2 \rfloor\}, \qquad S_{j+1}(n) := \{ n \bmod k : k \in S_j(n) \setminus\{0\} \}, \qquad j \geq 0.$$ Set $s_j(n) = |S_j(n)|$, with $s_1(n) = s(n)$.

This structure directly relates to the maximal length in Pierce expansions. Consider the process: for $a_0 \in [1, n]$, recursively define $a_{i+1} = n \bmod a_i$ until reaching zero. The quantity $P(n)$ is the maximal number of steps needed as $a_0$ ranges over $[\lfloor n/2 \rfloor + 1, n]$. For $j \geq 1$, $r \in S_j(n)$ if there exists $a$ such that the $(j+1)$-st step in the Pierce process yields $r$. Thus, $S_j(n)$ encodes all possible remainders at depth $j$ across all maximal initial $a$.

The sizes of iterated remainder sets satisfy tight uniform bounds: $$\frac{1}{(j+2)!} \leq \liminf_{n \to \infty} \frac{s_j(n)}{n} \leq \limsup_{n \to \infty} \frac{s_j(n)}{n} \leq \frac{1}{j+2}$$ for each fixed $j$. This establishes that although $s_j(n)$ is linearly large in $n$, the proportionality decreases factorially with $j$.

5. Context and Relation to Integer Sequence Taxonomy

The arithmetic and combinatorial structure of A283190 situates it among a broad class of integer sequences where a simple local rule (here, modular reduction over a specified $k$–range) leads to intricate global (asymptotic and local) behavior. Connections arise with:

• Aliquot sequences and the residue structure of iterated divisor sums, though the statistics and empirical catalogs in that domain address different but related questions (Bosma, 2016).
• Variants of Stern’s diatomic sequence, where recurrences are perturbed (e.g., via non-linear operations, alternative indexing, or counting structures other than the binary digit weight) and which exhibit analogous phenomena such as non-monotonicity, block patterns, or recursive self-similarity (Northshield, 2015).
• The paper of lexicographically earliest or “greedy” sequences constrained by set or word patterns in the OEIS (Sloane, 2018), though A283190’s definition is strictly modular–arithmetic.

Interest in A283190 is particularly motivated by how small changes in the definition—such as the range of $k$ for modular reduction or the recursive application in iterated sets—give rise to quantitatively distinct but structurally comparable sequences.

6. Analytical Techniques and Open Questions

The results for A283190 draw upon sieve theory, combinatorial decomposition, and the analysis of remainder-transfer under sequential increment. The precise value of the main constant $c$ comes from a product–sum over primes, reminiscent of constants in the distribution of reduced residues.

While the main asymptotic is sharp, several finer questions remain:

• The possible limiting distribution of $s_j(n)/n$ for higher $j$, given the numerically observed oscillatory behavior and the absence of convergence for $j \geq 2$.
• The explicit characterization of the set of $n$ where sharp drops in $s(n+1)-s(n)$ occur and the detailed structure of those $n$ producing maximal (or minimal) remainder diversity.
• Structural analogies to similar sequences in continued fraction or integer expansion (such as those addressed by generalized Stern-type sequences) remain a fertile ground for future research.

7. Summary Table of Key Properties

Property	Formula/Statement	Comments
Definition	$$s(n) = |\{n \bmod k : 1 \leq k \leq \lfloor n/2 \rfloor\}|$$	OEIS A283190
Growth	$$s(n) = c n + O\left(\frac{n}{\log n \log\log n}\right)$$	$c \approx 0.2296$; explicit product–sum over primes
Maximum jump	$s(n+1) \leq s(n) + 1$	No jump >1
Arbitrary drops	$\liminf_{n \to \infty} [s(n+1)-s(n)] = -\infty$	Dips of arbitrary size exist
Iterated set size bounds	$$\frac{1}{(j+2)!} \leq \liminf \frac{s_j(n)}{n} \leq \limsup \frac{s_j(n)}{n} \leq \frac{1}{j+2}$$	For $j \geq 0$; tight up to constants

Sequence A283190 thus serves as a natural and illuminating case paper in the distribution of modular reductions, linking elementary arithmetic operations to subtle global properties, and, via its generalizations, to broader phenomena in integer sequence theory, iterative processes, and number-theoretic expansions (Baraskar et al., 28 Aug 2025).