The Untouchables

Pyth, 21 bytes

.f!fqZsf!%TYStTSh^Z2Q

Warning: Incredibly slow. Test run and timing below.

$ time pyth -c '.f!fqZsf!%TYStTSh^Z2Q' <<< 3
[2, 5, 52]

real    2m46.463s
user    2m46.579s
sys 0m0.004s

It's basically as brute force as possible, tests factorizations up to the potential lonely number squared plus one.

C, 104 bytes

j,i,z,w;u(y){for(z=2;y;z++)for(w=0;(++w<z*z||0&printf("%i ",z)+y--)&&j^z;)for(i=j=0;++i<w;j+=i*!(w%i));}

It will take a few minutes for y > 20, but whatever.

Java, 310 Bytes

class U{int[] p;U(int e){p=new int[(int)4e9];for(int i=1,c=0;c<e;i++)if(u(i)>0){System.out.println(i);c++;}}int p(int n){if(p[n]!=0)return p[n];int i,s=1;for(i=2;i<=n-1;i++)s+=n%i==0?i+(n/i!=i?n/i:0):0;return(p[n]=s);}int u(int n){if(n==1)return 0;for(int i=2;i<=(n-1)*(n-1);i++)if(p(i)==n)return 0;return 1;}}

Golfed as well as I could but, I was more interested in making sure it ran in reasonable time. The unglofed version is probably more interesting

public class Untouchable {
    int[] properDivisorSumMap;
 
    
    public Untouchable(int estimatedMaxium){
        properDivisorSumMap = new int[(estimatedMaxium-1)*(estimatedMaxium-1)];
    }
    
    
    public int properDivisorSum(int n){
        if(properDivisorSumMap[n] != 0){
            return properDivisorSumMap[n];
        }
        
        int sum = 1;
        for(int i=2;i<=(int)Math.sqrt(n);i++){
            if(n%i==0){
                sum+=i;
                if(n/i != i){
                    sum+=n/i;
                }
            }
        }
        properDivisorSumMap[n] = sum;
        return sum;
    }
    
    
    public boolean untouchable(int n){
        if(n==1){
            return false;
        }
        for(int i=2;i<=(n-1)*(n-1);i++){
            if(properDivisorSum(i) == n){
                return false;
            }
        } 
        return true;
    }

    
    public static void main(String[] args){
        Untouchable test = new Untouchable(8480);
        
        int elements = Integer.parseInt(args[0]);
        
        for(int i=1,count=0;count < elements;i++){
            if(test.untouchable(i)){
                System.out.printf("%4d: %4d%n",count,i);
                count++;
            }
        }
    }
}