On the prime power factorization of n (original) (raw)

On the Parity of Exponents in the Factorization ofn!

Journal of Number Theory, 1997

It is shown that, for any k, there exist infinitely many positive integers n such that in the prime power factorization of n!, all first k primes appear to even exponents. This answers a question of Erdo s and Graham (``Old and New Problems and Results in Combinatorial Number Theory,'' L'Enseignement Mathe matique, Imprimerie Kundia, Geneva, 1980). A few generalizations are provided as well. 1997 Academic Press Question. Does there exist, for every fixed k, some n>1 with all the exponents _ 1 (n), _ 2 (n), ..., _ k (n) even? Our first result answers this question in the affirmative.

ON FACTORIZATION OF INTEGERS WITH RESTRICTIONS ON THE EXPONENTS

2007

Abstract For a fixed k∈ N we consider a multiplicative basis in N such that every n∈ N has the unique factorization as product of elements from the basis with the exponents not exceeding k. We introduce the notion of k-multiplicativity of arithmetic functions, and study several arithmetic functions naturally defined in k-arithmetics. We study a generalized Euler function and prove analogs of the Wirsing and Delange theorems for k-arithmetics.

On the Number of Factorizations of an Integer

Integers, 2011

Let f (n) denote the number of unordered factorizations of a positive integer n into factors larger than 1. We show that the number of distinct values of f (n), less than or equal to x, is at most exp C log x log log x (1 + o(1)) , where C = 2π 2/3 and x is sufficiently large. This improves upon a previous result of the first author and F. Luca.

On the decomposition of n! into prime powers

Journal of Number Theory, 1977

ABSTRACT If n is a positive integer, we write n! as a product of n prime powers, each at least as large as nδ(n). We define α(n) to be max δ(n), where the maximum is taken over all decompositions of the required type. We then show that limn→∞α(n) exists, and we calculate its value.

A characterization of nonprime powers

TURKISH JOURNAL OF MATHEMATICS, 2017

A criterion is presented in order to decide whether a given integer is a prime power or not. The criterion associates to each positive integer m a finite set of integers S(m) , each of them < m , and the properties of this set are studied. The notion of complementary pairs in S(m) is introduced and it is proved that if one is able to determine a complementary pair n, n ′ , then a partial factorization of the odd integer m can be obtained in polynomial time. Some particular cases and examples of these results are given.

Some conjectures in elementary number theory, II

2014

We announce a number of conjectures associated with and arising from a study of primes and irrationals in R. All are supported by numerical verification to the extent possible. This is an unpublished updated version as of August 13, 2014.

On the decomposition of n! into primes

Abstract. In this note, we make explicit approximation of the average of prime powers in the decomposition of n!. Then we find the order of geometric and harmonic means of such powers. ... 1.1. Approximate Formula for the Function Υ(n). First, we note that integrating by

Two Remarks on the Largest Prime Factors of n and n + 1

2020

Let P (n) be the largest prime factor of n. We give an alternative proof of the existence of infinitely many n such that P (n) > P (n + 1) > P (n + 2). Further, we prove that the set {P (n+1)/P (n)}n∈N has infinitely many limit points {0, xn, 1, yn}n∈N with 0 < xn < 1 < yn and limxn = lim yn = 1.