Number theory question

[raw elm]

This is something i am just curios about. I had observed this a few months ago but could not make progress. The problem is that to find a general form of square numbers whose sum of their divisors are also square numbers. An example of this would be 81. The sum of 81's divisors is 121 which is a square.

[broken chasm]

Well there is a function that provides the sum of divisors in number theory the sigma function
We can take a gen form of a square no and equate it with the sum of divisors formula

[raw elm]

So you are saying :-
Let $n = {p_1}^{d_1} {p_2}^{d_2} ................ {p_k}^{d_k}$.
So we can write $\sigma(n^2) = \sigma ({p_1}^{2d_1} {p_2}^{2d_2} ……………. {p_k}^{2d_k})$

[median karma]

Someone_Random

[broken chasm]

I mean dont jump right into the general case take some examples of numbers satisfying your argument try to identify a pattern in those and then apply that pattern in genral case

[raw elm]

oh ok

[raw elm]

I dont seem to find a pattern in the numbers that satisfy this condition

a sequncefrom this series is 1 , 81 ,400 , 32400 , 1705636 ,3648100

or 1 , 9 , 20 , 180 , 1306 , 1910

[raw elm]

?

[broken chasm]

Hmmmm

[raw elm]

I cannot see a pattern

[broken chasm]

My thought now is to remove the original condition that the sum of numbers not just square numbers is a perfect square il check numbers and try to devise something and update you then we can add the original condition and see what new happens

[raw elm]

oh good idea

you can go to https://oeis.org/A006532 and check the sequnce out

[broken chasm]

Rn I'm unable to identify a pattern il advice you to ask the seniors here they'll guide you better ig

[raw elm]

sure

[wild vigil]

This problem feels open

You're looking for all squares of the form
[\prod_{i=1}^k\left(1+p_i+p_i^2+\dots+p_i^{2a_i}\right)]
where $p_1,p_2,\dots,p_k$ are distinct primes and $a_1,a_2,\dots,a_k$ are positive integers.

[median karma]

Daniel

[wild vigil]

That seems intractable

[raw elm]

what do you mean?

[wild vigil]

The problem is very hard, probably unsolved

[raw elm]

tru

[lavish kelp]

.close