## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  488.10045
Autor:  Erdös, Paul; Nicolas, Jean-Louis
Title:  Grandes valeurs d'une fonction liée au produit d'entiers consecutifs. (Large values of a function related to the product of consecutive integers.) (In French)
Source:  Ann. Fac. Sci. Toulouse, V. Ser., Math. 3, 173-199 (1981).
Review:  Define f(n) to be the largest integer k for which there is an m such that n|(m+1)...(m+k) but n\not|(m+j) for 1 \leq j \leq k. The authors prove that sumn \leq xf(n) ~ x log log x and that maxn \leq xf(n) = \frac{e\gamma/2 log x}{2(log log x) ½}+\frac{\gamma e\gamma log x}{4 log log x}(1+o(1)), where \gamma is Euler's constant. Define n to be f-highly aboundant if for every n' < n, f(n') < f(n). The authors make a detailed study of the prime factors of f-highly aboundant numbers. The results show similarity with the highly composite numbers of Ramanujan, but there are some differences. For example, there are large gaps in the factorization of f-highly aboundant numbers.
Reviewer:  S.W..Graham
Classif.:  * 11N37 Asymptotic results on arithmetic functions
11N05 Distribution of primes
Keywords:  large values of functions; product of consecutive integers; prime factors of f-highly abundant numbers

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