Unitary untouchable numbers

[Dean Hickerson]

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