Antichains among Divisor Sums of Divisors

Abstract

For a positive integer $n$, let $S(n)=\{\sigma (d):d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma (d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding product of divisibility posets, and this maximum is attained infinitely often. In the squarefree case with $\omega (n)=r$, every width from $1$ to $\left(\genfrac{}{}{0ex}{}{r}{\lfloor r/2\rfloor }\right)$ occurs, again infinitely often. For repeated exponent patterns, Engel's limit theory reduces the asymptotic width problem to optimal representation variance. We show that for the divisibility poset $D_m=(\{1,\dots ,m\},\mid )$, the prime directions $2$ and $3$ already determine the exact variance. The structural reason is the critical dilation $4$, lying strictly between $3$ and the next prime $5$. This yields an exact finite formula, the limit $V(D_m)\to 1.0367934201598\dots$, and asymptotics for repeated exponent patterns. The complementary minimum-width problem remains largely open.

arXiv comment20 pages, 2 figures