  We study a bijective map from integer partitions to the prime factorizations
of integers that we call the "supernorm" of a partition, in which the
multiplicities of the parts of partitions are mapped to the multiplicities of
prime factors of natural numbers. The supernorm is connected to a family of
maps we define, which suggests the potential to apply techniques from partition
theory to identify and prove multiplicative properties of integers. We make a
brief study of pertinent analytic aspects of the supernorm. Then, as an
application of "supernorma"' mappings (i.e., pertaining to the supernorm
statistic), we prove an analogue of a formula of Kural-McDonald-Sah to give
arithmetic densities of subsets of $\mathbb N$ instead of natural densities in
$\mathbb P$ like previous formulas of this type; this builds on works of
Alladi, Ono, Wagner, and the first and third authors. Finally, using a table of
"supernormal" additive-multiplicative correspondences, we conjecture
Abelian-type formulas that specialize to our main theorem and other known
results.
