Unitary untouchable numbers
Dean Hickerson
dean at math.ucdavis.edu
Tue Sep 4 03:20:42 CEST 2001
felice.russo at katamail.com asked:
> I would like to extend the concept of untouchable numbers to the unitary
> divisors. So the definiton of unitary untouchable numbers should be:
>
> The numbers n such that us(x)=n has no solution where us(x) is the sum of
> unitary proper divisors of x.
>
> With a ubasic code for n<=10^5 I have found the following ones:
>
> 2 3 4 5 7 374 702 758 926 930 998
>
> But how I can sure that none of those became solution of us(x)=n for
> larger values of n (>10^5)?
>
> Is there any theoretical approach to prove that a number is unitary
> untouchable?
Suppose that us(x) = n > 1. Then x has at least one unitary divisor that's
not equal to either 1 or x. (Otherwise us(x) would be 1.) Let d be such a
divisor. Then x/d is also a unitary divisor of x that's not equal to 1 or
x. Since d and x/d are relatively prime, they can't be equal. Hence d and
x/d are distinct proper unitary divisors of x, so
d + x/d <= us(x) = n.
Multiplying by 4d and rearranging gives
(2d - n)^2 <= n^2 - 4x,
so
n^2 - 4x >= 0.
Therefore, given n, you only need to check the values of x up to n^2/4.
For the numbers you listed, I find that us(102016) = 926 (as Richard Guy
mentioned), and us(172288) = 930. The others are unitary untouchable.
Dean Hickerson
dean at math.ucdavis.edu
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