Unusual numbers

Simon Colton simonco at cs.york.ac.uk
Tue Aug 28 16:37:54 CEST 2001

Steven, Neil,

>Let's therefore bring this story to an end.
>Please call A063538 "sqrt(n)-rough" and
>A063539 "non-sqrt(n)-rough".
>Remember that a positive integer is called
>y-smooth if all its prime factors are <= y.
>Hence, to deal correctly with the equality
>case, I've introduced the new term "rough":
>a positive integer is called y-rough if all
>its prime factors are >= y.

Here's a little result about A063539:

Given any two integers k and n such that n^2 > k > 1, then:

kn(n+1) will be non-sqrt(n)-rough

[i.e. the largest prime factor of kn(n+1) will be less than the
square root of kn(n+1)].

The proof is straighforward, relying on the obvious fact that
lpf(x) =< x [where lpf is the largest prime factor] and the fact
that lpf(k(n(n+1))) will either be lpf(k), lpf(n) or lpf(n+1).

The question is: is there a more general result of this nature?

As usual with me, this conjecture came about by using the
NumbersWithNames program available to run here:
(I generalised three of the conjectures it made).

All the best,


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