Let us say that a positive integer is factor-rich if it has more factors than any smaller integer. For example, $60$, which has twelve factors, is factor-rich; and therefore $72$, which also has twelve factors, is not. The first few factorials are factor-rich. However, $8!=2^73^25\cdot7$ has $8\cdot3\cdot2\cdot2=96$ factors, while $6\cdot7!$ has no fewer; so $8!$ is not factor-rich.

Suppose that one is looking for big factor-rich numbers. For such a number $N$, one would expect its prime-power factors to comprise a high power of $2$, a somewhat lesser power of $3$, a considerably smaller power of $5$, and so on, until a tail of single prime factors is reached. As $N$ increases, one might conjecture that the relative proportions of the indices stabilize. Namely, if $N$ is factor-rich with $N=\prod_{k=1}^np_k^{m_k}$, where $p_k$ is the $k$th prime and $m_1,...,m_n$ are positive integers, the conjecture is that $$\lim_{N\,\text{large}}\frac{m_k}{m_{k+1}}$$ is well defined for the first few $k$, and actually equals $1$ for the last few (the tail), while the ratio $m_k/m_{k+1}$ behaves more erratically for some mid-range $k$ near the tail.

Is this conjecture right? If so, what is a more precise statement of it? Or, if not, what is known about the distribution of the ratios $m_k/m_{k+1}$ as $N$ increases?


1 Answer 1


You'll be helped in Ramanujan's paper http://ramanujan.sirinudi.org/Volumes/published/ram15.pdf


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