[seqfan] Re: A conjeture about 34155

[William Rex Marshall]

On 15/04/2012 6:05 a.m., Claudio Meller wrote:
> Hi seqfans, one of my blog readers, Jordi Domènech i Arnau, has sent me
> this :
>
> *I have looked for the numbers that are equal to the sum of their proper
>> divisors, greater than their square root. *
>
> *I found a lot of even numbers with this property, but  I´ve only found
>>   one odd number, 34155, with this property.*
>
>
>
> He makes the conjeture that 34155 is the only odd number with this
> caracteristic.
>
> 34155 = 207+253+297+345+495+621+759+1035+1265+1485+2277+3105+3795+6831+11385
>
> I ´ve made the sequence (not in the OEIS):
> Numbers N  equal to the sum of its proper divisors greater than sqr (N)

I would prefer including the square root of N in the sum, which is of 
course possible when N is a square. Then it could also be defined as 
numbers N for which the sum of the long sides of integer rectangles of 
area N equals twice N. 34155 is still the only known odd number in this 
case.

It doesn't seem to result in a different sequence though, or does it? 
Does a square number exist which is satisfied by one definition but not 
the other?