## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  513.10043
Autor:  Canfield, E.R.; Erdös, Paul; Pomerance, Carl
Title:  On a problem of Oppenheim concerning "Factorisatio Numerorum". (In English)
Source:  J. Number Theory 17, 1-28 (1983).
Review:  Denote by f(n) the number of factorizations of a positive integer n into factors exceeding 1, the order of the factors being immaterial. In this interesting paper, the authors establish a good estimate for the maximal order of f(n), thus correcting a result of A.Oppenheim [J. Lond. Math. Soc. 1, 205-211 (1926); ibid. 2, 123-130 (1927)]; their estimate is of the form

n\exp(- log n(log n)-1 log3n(1+E(n)))

where E(n) = o(1) as n ––> oo and is given rather more explicitly in the paper, and where logk n denotes the k-fold iterated logarithm. A new lower bound for \Psi(x,x1/u), the number of positive integers n \leq x with no prime factor exceeding x1/u, is also derived (and applied), namely

\Psi(x,x1/u) \geq x\exp(-u(log u+(log2u-1)(1+\frac{1}{log u})+..elke..(log22u log-2u)))

for x \geq 1, u \geq 3. The paper concludes with an investigation of the largest prime divisors of highly factorable numbers n, i.e. those n for which f(m) < f(n) whenever m < n (in which case f(n) has maximal order). The 118 highly factorable numbers up to 109 are listed, and the algorithm used to obtain them described. Some additional questions are raised.
Reviewer:  E.J.Scourfield
Classif.:  * 11N37 Asymptotic results on arithmetic functions
11N05 Distribution of primes
11B83 Special sequences of integers and polynomials
11A25 Arithmetic functions, etc.
Keywords:  number of factorizations of positive integer; maximal order; lower bound for Psi-function; largest prime divisors of highly factorable numbers

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