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

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Sungjin Kim

Department of Mathematics

Santa Monica College

California State University, Northridge

18111 Nordhoff Street

Northridge, CA 91330

USA

**Abstract:**

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, *x*_{n}, 1, *y*_{n}}_{n∈N}
with 0 < *x*_{n} < 1 < *y*_{n} and
lim *x*_{n} = lim *y*_{n} = 1.

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(Concerned with sequence
A006530.)

Received June 15 2020;
revised versions received September 7 2020; October 16 2020.
Published in *Journal of Integer Sequences*,
October 16 2020.

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