Numbered Triangle: (lack of)Prime Patterns

My first post here, Hi All.
Assume we arrange numbers like this.

   
                   1
                  2 3
                4  5  6 
              7  8  9 10
            11 12 13 14 15
          16 17 18 19 20 21
         22 23 24 25 26 27 28
       29 30 31 32 33 34 35 36
      37 38 39 40 41 42 43 44 45
    46 47 48 49 50 51 52 53 54 55
   56 57 58 59 60 61 62 63 64 65 66
 67 68 69 70 71 72 73 74 75 76 77 78
79 80 81 82 83 84 85 86 87 88 89 90 91

There appears to be a pattern between diagonal columns that can contain prime numbers to those that never will. The 2,3,5,7 are anomalies because no primes exist past these numbers. I created a script to quickly check for primes down to the billionth row to satisfy my definition of no primes in these diagonal rows.

1 3-never 0 prime rows in between
2 5-never 1 prime rows in between
4 8-primes
7 12-never 2 prime rows in between
11 17-primes
16 23-primes
22 30-never 3 prime rows in between
29 38-primes
37 47-primes
46 57-primes
56 68-never 4 prime rows in between
67 80-primes
79 93-primes
92 107-primes
106 122-primes
121 138-never 5 prime rows in between

The gap between a diagonal column never containing a prime starts at 0 and increases by one. I've tested this for a few thousand rows and it holds true.

My question is why?
What is happening here?

Obviously I assume this has to do with ordering the numbers in such a fashion. But for the life of me I cannot understand why this is happening.

Any nudge in the right direction would be appreciated.

Thanks

[wblipp]

Originally Posted by Stanek

Any nudge in the right direction would be appreciated.

Find the quadratic expression for each of your diagonals (hint - the far right diagonal is n(n+1)/2, the others are offset from this.)

Then try factoring the expression with the quadratic formula.

William