Through numerical calculations I have discovered that for any natural number $n \geq 2$, the geometric mean of the prime factors of all natural numbers $\leq n$ can be approximated well by $1.6653 \cdot n^{0.224}$. Is there a number-theoretic explanation for why this is a good approximation?

Let $(C,\otimes)$ be a monoidal (non symmetric) Waldhausen category. In particular, under these conditions, $K_{0}(C)$ is a ring and $K_{i}(C)$ are $K_{0}(C)$-bimodule for any $i\in \mathbb{Z}$. Suppose that $K_{0}(C)$ is a commutative ring. Let $a\in K_{i}(C)$ and $q\in K_{0}(C) $. I was wonderi...

An amorphous set is an infinite set that cannot be partitioned into 2 infinite subsets. An amorphous cardinality is the cardinality of an amorphous set. Working in $\sf ZF$, it is consistent that amorphous sets exist. Amorphous sets necessarily don't have a lot of structure. For example, for an a...

The short version of my question is: Suppose $X$ is a (reasonably nice) topological space such that $X = \bigcup_{n \ge 1} X_n$ for an increasing sequence of (closed) subspaces $X_1 \subset X_2 \subset \cdots$​, and suppose that $\mathcal{F}$ and $\mathcal{G}$ are sheaves on $X$ which are isomorp...