On a Problem of Oppenheim concerning “ Factorisatio Numerorum ”
نویسنده
چکیده
Letf(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)-" " " ' 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 thatf(n) = n 'L(n)-'+ " " '. Some results on the multiplicative structure of highly factorable numbers are proved and a table of them up to lo9 is provided. Of independent interest, a new lower bound is established for the function Y(x, y), the number of n <x free of prime factors exceeding y.
منابع مشابه
On Some Asymptotic Formulas in the Theory of the "factorisatio Numerorum"
ON SOME ASYMPTOTIC FORMULAS IN THE THEORY OF THE "FACTORISATIO NUMERORUM" BY P. ERDÖS (Received December 2, 1940) Let 1 < a, < a2 < . . . be a sequence of integers . Denote by f (n) the number of representations of n as the product of the a's, where two representations are considered equal only if they contain the same factors in the same order . As far as I know the first papers written on the...
متن کاملDistribution of the number of factors in random ordered factorizations of integers
We study in detail the asymptotic behavior of the number of ordered factorizations with a given number of factors. Asymptotic formulae are derived for almost all possible values of interest. In particular, the distribution of the number of factors is asymptotically normal. Also we improve the error term in Kalmár’s problem of “factorisatio numerorum” and investigate the average number of distin...
متن کاملOn the maximal order of numbers in the “factorisatio numerorum” problem
Let m(n) be the number of ordered factorizations of n ≥ 1 in factors larger than 1. We prove that for every ε > 0 m(n) < nρ exp ( (log n)1/ρ/(log log n)1+ε ) holds for all integers n > n0, while, for a constant c > 0, m(n) > nρ exp ( c(log n/ log log n)1/ρ ) holds for infinitely many positive integers n, where ρ = 1.72864 . . . is the real solution to ζ(ρ) = 2. We investigate also arithmetic pr...
متن کاملar X iv : m at h / 05 10 05 4 v 2 [ m at h . H O ] 1 7 A ug 2 00 6 EULER AND THE PENTAGONAL NUMBER THEOREM
In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem to the divisor function, partition function and divergent series. We have attempted to give an exhaustive review of all of Euler's correspondence and publications about the pentagonal number theorem and his applications of it. Comprehensus: In hoc di...
متن کاملA Generalization of a Conjecture of Hardy and Littlewood to Algebraic Number Fields
We generalize conjectures of Hardy and Littlewood concerning the density of twin primes and k-tuples of primes to arbitrary algebraic number fields. In one of their great Partitio Numerorum papers [7], Hardy and Littlewood advance a number of conjectures involving the density of pairs and k-tuples of primes separated by fixed gaps. For example, if d is even, we define Pd(x) = |{0 < n < x : n, n...
متن کامل