Prime Factors of Consecutive Integers

1972 ◽  
Vol 79 (10) ◽  
pp. 1082-1089 ◽  
Author(s):  
E. F. Ecklund ◽  
R. B. Eggleton
Keyword(s):  
Prime Factors ◽  
Consecutive Integers

On triplets with descending largest prime factors

2001 ◽  
Vol 38 (1-4) ◽  
pp. 45-50 ◽  
Author(s):  
A. Balog
Keyword(s):  
Prime Factor ◽  
Prime Factors ◽  
Consecutive Integers

For an integer n≯1 letP(n) be the largest prime factor of n. We prove that there are infinitely many triplets of consecutive integers with descending largest prime factors, that is P(n - 1) ≯P(n)≯P(n+1) occurs for infinitely many integers n.

scholarly journals On strings of consecutive integers with no large prime factors

1998 ◽  
Vol 64 (2) ◽  
pp. 266-276 ◽  
Author(s):  
Antal Balog ◽  
Trevor D. Wooley
Keyword(s):  
Integer Coefficients ◽  
Prime Factors ◽  
Consecutive Integers ◽  
Binomial Polynomials

AbstractWe investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ε, we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding nε, with the length of the strings tending to infinity with speed log log log log n.

Multiplicative functions at consecutive integers

1986 ◽  
Vol 100 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Random Sequence ◽  
Multiplicative Functions ◽  
Liouville Function ◽  
Prime Factors ◽  
Consecutive Integers ◽  
Image Position

Let λ(n) denote the Liouville function, i.e. λ(n) = 1 if n has an even number of prime factors, and λ(n) = − 1 otherwise. It is natural to expect that the sequence λ(n) (n ≥ 1) behaves like a random sequence of ± signs. In particular, it seems highly plausible that for any choice of εi = ± 1 (i = 0,…, k) we have

scholarly journals On the smooth values of shifted almost-primes

2019 ◽  
Vol 15 (01) ◽  
pp. 1-9
Author(s):  
Xiaodong Lü ◽  
Zhiwei Wang ◽  
Bin Chen
Keyword(s):  
Lower Bound ◽  
Prime Factor ◽  
Almost Primes ◽  
Prime Factors ◽  
Consecutive Integers

Denote by [Formula: see text] (respectively, [Formula: see text]) the largest (respectively, the smallest) prime factor of the integer [Formula: see text]. In this paper, we prove a lower bound of almost-primes [Formula: see text] with [Formula: see text] such that [Formula: see text] for [Formula: see text]. As an application, we study two patterns on the largest prime factors of consecutive integers with one of which without small prime factor.

scholarly journals Consecutive integers with no large prime factors

1976 ◽  
Vol 22 (1) ◽  
pp. 1-11 ◽  
Author(s):  
R. B. Eggleton ◽  
J. L. Selfridge
Keyword(s):  
Lower Bounds ◽  
Prime Factor ◽  
Typical Result ◽  
Prime Factors ◽  
Consecutive Integers

For fixed integers k and m, with k ≥m ≥ 2, there are only finitely many runs of m consecutive integers with no prime factor exceeding k. We obatin lower bounds for the last such run. Let g(k, m) be its smallest member. For 2 ≤ m ≤ 5 it is shown that g(k, m) > kc logloglogk holds for all sufficiently large k, where c is a constant depending only on m. We also obtain a number of lower bounds with explict ranges of validity. A typical result of this type g(k, 3) > k3 holds just if k ≥ 41.

scholarly journals Prime factors of consecutive integers

2008 ◽  
Vol 77 (264) ◽  
pp. 2455-2459 ◽  
Author(s):  
Mark Bauer ◽  
Michael A. Bennett
Keyword(s):  
Prime Factors ◽  
Consecutive Integers

scholarly journals On strings of consecutive integers with a distinct number of prime factors

2008 ◽  
Vol 137 (05) ◽  
pp. 1585-1592 ◽  
Author(s):  
Jean-Marie De Koninck ◽  
John B. Friedlander ◽  
Florian Luca
Keyword(s):  
Prime Factors ◽  
Consecutive Integers
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