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**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} log_{3}n(1+E(n))) where E(n) = o(1) as n ––> oo and is given rather more explicitly in the paper, and where log_{k} n denotes the k-fold iterated logarithm. A new lower bound for \Psi(x,x^{1/u}), the number of positive integers n \leq x with no prime factor exceeding x^{1/u}, is also derived (and applied), namely

\Psi(x,x^{1/u}) \geq x\exp(-u(log u+(log_{2}u-1)**(**1+\frac{1}{log u}**)**+..elke..(log_{2}^{2}u log^{-2}u))) 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 10^{9} 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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag