What the Least Common Multiple Divides, II

✍ Scribed by G. Myerson; J.W. Sander

• Book ID: 102974227
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 504 KB
• Volume: 61
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0138

✦ Synopsis

In Part I, we defined an arithmetical function whose values are certain multinomial coefficients. Here, we study the rate of growth and divisibility properties of the function. No familiarity with Part I is assumed here.

1996 Academic Press, Inc.

1. BACKGROUND, PROBLEMS, MAIN RESULTS

For real x>1, let L(x) be the least common multiple, and P(x) the product, of the numbers 1, 2, ..., [x]. In an earlier paper [2], one of us proved that, for all natural n, L(n) divides P(n) P(nÂ2) P(nÂ3) P(nÂ7) P(nÂ43) } } } , where the sequence 2, 3, 7, 43, ... is the one in which each term is one greater than the product of all the preceding terms. Accordingly, he defined the integer-valued function f (n) by

He raised the question of the rate of growth of f (n), pointing out that, for small n, the behavior of f is quite erratic, e.g., f (95)=542640, f (114)=3.

He also raised the question, whether f (n) was odd infinitely often. This was article no.