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Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the sums of $F(n)^{beta}$ and $G(n)^{beta}$ up to $x$ for all real $beta$ and the asymptotic bounds for $f(n)^{beta}$ and $g(n)^{beta}$ for all negative $beta$.
We investigate the integer solutions of Diophantine equations related to perfect numbers. These solutions generalize the example, found by Descartes in 1638, of an odd, ``spoof perfect factorization $3^2cdot 7^2cdot 11^2cdot 13^2cdot 22021^1$. More r
In this article we study the norm of an integer partition, which we define to be the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of current rese
** The primary topic of this dissertation is the study of the relationships between parts and wholes as described by particular physical theories, namely generalized probability theories in a quasi-classical physics framework and non-relativistic qua
We show that for $n geq 3, n e 5$, in any partition of $mathcal{P}(n)$, the set of all subsets of $[n]={1,2,dots,n}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,Csubseteq [n]$ such that $Acap B$, $Acap
We introduce a bijection between inequivalent minimal factorizations of the n-cycle (1 2 ... n) into a product of smaller cycles of given length, on one side, and trees of a certain structure on the other. We use this bijection to count the factoriza