On Highly Factorable Numbers

✍ Scribed by Jun Kyo Kim

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 287 KB
• Volume: 72
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2238

✦ Synopsis

For a positive integer n, let f (n) be the number of multiplicative partions of n. We say that a nutural number n is highly factorable if f (m)< f (n) for all m, 1 ma j then f (np j Âp i ) f (n). Using this fact, we prove the conjecture of Canfield, Erdo s, and Pormerance: for each fixed k, if n is a large highly factorable number then there are asymptotically exactly 1Âk(k+1) of the exponents of n which are equal to k. We also answer the questions posed by Canfield et al.: if n, n$ are consecutive highly factorable numbers, then does it follow n$Ân Ä 1 and f (n$)Âf (n) Ä 1?