On a problem of Oppenheim concerning “factorisatio numerorum”

  title={On a problem of Oppenheim concerning “factorisatio numerorum”},
  author={E. Rodney Canfield and Paul Erd{\"o}s and Carl Pomerance},
  journal={Journal of Number Theory},

Tables from this paper

On Highly Factorable Numbers

Abstract 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⩽ m n . We show

"Factorisatio numerorum" in arithmetical semigroups

Introduction. The problems of “factorisatio numerorum”, which go back more than 65 years, are concerned principally with (i) the total number f(n) of factorizations of a natural number n > 1 into

On ordered factorizations into distinct parts

Let g(n) denote the number of ordered factorizations of n into integers larger than 1. In the 1930s, Kalmár and Hille investigated the average and maximal orders of g(n). In this note we examine

Factoring Integers and Computing Discrete Logarithms via Diophantine Approximations

  • C. Schnorr
  • Mathematics, Computer Science
  • 1991
It is shown, under the assumption that the smooth integers distribute "uniformly", that there are Ne+o(1) many solutions (e1,...,et) if c > 1 and if Ɛ := c - 1 - (2c - 1) log log N / log pt > 0.

On the ultimate complexity of factorials

  • Qi Cheng
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 2003

Factoring with an Oracle

  • U. Maurer
  • Mathematics, Computer Science
  • 1992
A polynomial-time oracle factoring algorithm for general integers is presented which asks at most Ɛn oracle questions for sufficiently large N, and it is shown that the algorithm fails with probability at most N-Ɛ/2 for all sufficientlylarge N.

A rigorous time bound for factoring integers

In this paper a probabilistic algorithm is exhibited that factors any positive integer n into prime factors in expected time at most Ln[2, 1 + o()] for n oo, where L,[a, b] = exp(b(logx)a(loglogx)l

Order computations in generic groups

It is proved that a generic algorithm can compute |α| for all α ∈ S ⊆ G in near linear time plus the cost of a single order computation with N = λ(S), and it is shown that in most cases the structure of an abelian group G can be determined using an additional O (Nδ/4 ) group operations, given an O ( Nδ ) bound on |G| = N.

The least common multiple of sets of positive integers

Motivated by the problem of estimating log lcm{f(k) : 1 ≤ f(k) ≤ n} when f is a polynomial, we study the typical behavior of the logarithm of the least common multiple of sets of integers in {1, . .



An asymptotic formula for extended Eulerian numbers

w,here H(n) H(n, ) k-l(k 1) __o d(n). The numbers H(n) are the extended Eulerian numbers; when n is square-free, H(n) is an Eulerian number. Properties of the extended Eulerian numbers may be found

On Highly Composite Numbers

for a certain c. In fact I shall prove that if n is highly composite, then the next highly composite number is less than n+n(log y&)-C ; and the result just stated follows immediately from this. At,

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

The Difference between Consecutive Prime Numbers V

  • R. Rankin
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 1963
Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n, where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4)

On the difference between consecutive prime numbers

/ = lim inf ̂ 11-tl . »->» log pn The purpose of this paper is to combine the methods used in two earlier papers1 in order to prove the following theorem. Theorem. (1) / = c(l + 40)/5, where c<

Corrections to Two of My Papers

In my paper " On th.e dit~ergence properties of th.e Lagran.ge interpolation poly-n.umiuZs, " (Annals of R4at. cos i7r (p and q odd), and the fundamental points of the interpolation are the roots of

The Enumeration of the Partitions of Multipartite Numbers

  • P. A. Macmahon
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1925
This paper is a study of a new method of enumeration of the partitions of multipartite numbers. Incidentally an algebraic function, which is derived from the repetitional exponents of partitions of