# On a problem of Oppenheim concerning “factorisatio numerorum”

@article{Canfield1983OnAP, 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}, year={1983}, volume={17}, pages={1-28} }

## 344 Citations

### On Highly Factorable Numbers

- Mathematics
- 1998

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

- Mathematics
- 1992

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

- Mathematics
- 2019

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

- Mathematics, Computer ScienceEUROCRYPT
- 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.

### Factoring with an Oracle

- Mathematics, Computer ScienceEUROCRYPT
- 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

- Mathematics
- 1992

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

- Mathematics, Computer Science
- 2007

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

- Mathematics
- 2011

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, . .…

## References

SHOWING 1-10 OF 21 REFERENCES

### An asymptotic formula for extended Eulerian numbers

- Mathematics
- 1974

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

- Mathematics
- 1944

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"

- Mathematics
- 1941

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

- MathematicsProceedings 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

- Mathematics
- 1975

/ = 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

- Mathematics
- 1943

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

- MathematicsMathematical 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…