+0  
 
+1
49
1
avatar+485 

If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \leq i \leq 2010$ and $f(i)=1+\sqrt{i}+i$?

 Apr 5, 2021
 #1
avatar+460 
+2

The LHS is an integer,  so the RHS must be as well and conversely $i=n^2$ for integer $n$.

 

Then sum of divisors of $n^2$ is $n^2+n+1$ then use the divisor sum formula.

 Apr 5, 2021

31 Online Users

avatar