On the maximal order of numbers in the "factorisatio numerorum" problem
Number Theory
2007-05-23 v2 Combinatorics
Abstract
Let m(n) be the number of ordered factorizations of n in factors larger than 1. We prove that for every eps>0 n^{rho} m(n) < exp[(log n)^{1/rho}/(loglog n)^{1+eps}] holds for all integers n>n_0, while, for a constant c>0, n^{rho} m(n) > exp[c(log n)^{1/\rho}/(loglog n)^{1/rho}] holds for infinitely many positive integers n, where rho=1.72864... is the real solution to zeta(rho)=2. We investigate also arithmetic properties of m(n) and the number of distinct values of m(n).
Keywords
Cite
@article{arxiv.math/0505352,
title = {On the maximal order of numbers in the "factorisatio numerorum" problem},
author = {Martin Klazar and Florian Luca},
journal= {arXiv preprint arXiv:math/0505352},
year = {2007}
}
Comments
We have rewritten the paper and improved considerably the lower bound. Thus now we know that the max. order of m(n) is n^{rho}/(exp((log n)^{1/rho+o(1)})). Submitted to JNT