Antichains among Divisor Sums of Divisors

For a positive integer $n$, let $S(n)=\{σ(d): d\mid n\}$ 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 $d\mapsto σ(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a cor