On a problem of Oppenheim concerning "factorisatio numerorum"

E. R. Canfield, Paul Erdös, Carl Pomerance

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183 Citations (Scopus)


Let f(n) denote the number of factorizations of the natural number n into factors larger than 1 where the order of the factors does not count. We say n is "highly factorable" if f(m)<f(n) for all m < n. We prove that f(n)=n·L(n)-1+0(1) for n highly factorable, where L(n)=exp{log n logloglog n loglog n}. This result corrects the 1926 paper of Oppenheim where it is asserted that f(n)=n·L(n)-2+0(1). Some results on the multiplicative structure of highly factorable numbers are proved and a table of them up to 109 is provided. Of independent interest, a new lower bound is established for the function Ψ(x, y), the number of n≤x free of prime factors exceeding y.

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalJournal of Number Theory
Issue number1
Publication statusPublished - Aug 1983

ASJC Scopus subject areas

  • Algebra and Number Theory

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