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I made this simple program to list all non-primes (ignore the first row and column of the output) and list what I call "important numbers". I have attached an output if you don't want to bother running and compiling the program.

Anyway, taking a look at the attached text file, I noticed that the "important numbers" (those which occurred 6 or more times in the grid) were more likely than a random number to have a prime before it or after it (at least for small numbers). After a Google search, I found out that a similar definition to my "important numbers" is given to "abundant numbers" and "super-abundant numbers" and "highly abundant numbers" -- that is they are determined by a high number of factors (for example, 12 = 1*12 = 2*6 = 3*4), just in a slightly different way.

I then began to wonder, is there a set of numbers that can be generated using a simple pattern such that every number in that set is one less than or one greater than a prime? Is there another set such that every prime is either one less than or one greater than a number in that set? Well, I haven't gotten far from this point.

I have found the set of numbers defined as: n(x) = 2((x^2)-x); where x is an integer greater than or equal to two, is more likely to have primes before or after its members, but that's about it.

Code:

```
#include <iostream>
#include <fstream>
using namespace std;
int main ()
{
ofstream output;
output.open ("output.txt");
int i;
int j;
int n;
cout<<"Size? ";
cin>>n;
int repeat[n*n];
for (j=1;j<=n;j++) {
for (i=1;i<=n;i++) {
output<<i*j<<"\t";
repeat[i*j]++;
}
output<<endl;
}
output<<endl<<endl<<endl<<"IMPORTANT NUMBERS:"<<endl;
for (i=1;i<=(n*n);i++) {
if (repeat[i]>=6) {
output<<i<<endl;
}
}
output.close();
return 0;
}
```

Anyway, taking a look at the attached text file, I noticed that the "important numbers" (those which occurred 6 or more times in the grid) were more likely than a random number to have a prime before it or after it (at least for small numbers). After a Google search, I found out that a similar definition to my "important numbers" is given to "abundant numbers" and "super-abundant numbers" and "highly abundant numbers" -- that is they are determined by a high number of factors (for example, 12 = 1*12 = 2*6 = 3*4), just in a slightly different way.

I then began to wonder, is there a set of numbers that can be generated using a simple pattern such that every number in that set is one less than or one greater than a prime? Is there another set such that every prime is either one less than or one greater than a number in that set? Well, I haven't gotten far from this point.

I have found the set of numbers defined as: n(x) = 2((x^2)-x); where x is an integer greater than or equal to two, is more likely to have primes before or after its members, but that's about it.