Oh the site is going slow but it did load. >Another cross reference took me off to sequence A002822, labeled “Numbers m such that 6m−1, 6m+1 are twin primes"

Oh the site is going slow but it did load. >Another cross reference took me off to sequence A002822, labeled “Numbers m such that 6m−1, 6m+1 are twin primes" Don't focus on 6m+/-1 alone and you'll get it. Let me explain All prime numbers above 2 are in the form of 2m + 1. ie. They are odd. So we have a formula that rules out half of numbers from being possible prime. Now we can do a similar thing and create a formula to rule out factors of 2 and 3. Factors of 2 and 3 repeat every 6 numbers (the multiple of 2 and 3). eg. 6m + 0, 6m + 2 or 6m + 4 is divisible by 2. 6m + 0 or 6m + 3 is divisible by 3. So all prime numbers above 3 are in the form of 6m + 1 or 6m + 5. We can write the 6m + 5 as 6m - 1 if we want. Only 2 out of 6 numbers can be prime above 3 and the numbers either side of these will have either a factor of 2 or a factor of 6. (Btw never simplify the 'only 2 out of 6 numbers can be prime' as there are windows (twin primes for example) where there's more than 1 our of 3 numbers that are prime. It's only true that there's never more than 2 out of 6 sequential numbers prime above 6. If you come up with a prime number theory that sets a maximum of the frequency of primes you need to consider the window size rather than a straight fraction or you'll end up with errors). Now we can do the same as above for 5. 30 is where factors of 2,3 and 5 align. All prime numbers above 30 are in the form of 30m + 1, 7, 11, 13, 17, 19, 23, 29. The rest are factors or 2,3 or 5. Only 8 out of 30 numbers above 5 can possibly be prime. All those primes are surrounded by a factor of 6 and 2 (since 6m +1/-1 is a subset of this) or in the +1/+29 case surrounded by a factor of 2 and a factor of 30. For the twin prime case 1/3 of prime numbers are next to multiples of 2,3,5 and 2/3 are next to multiples of 2,3. You could do the same with 2x3x5x7 = 210. And again come up with a formula for the maximum frequency of primes (48 out of 210 numbers above 7 can possible by prime) and also a formula for where primes can occur. In this case you'd see for the twin prime case 1/21 are next to 2x3x5x7, 6/21 next to 2x3x5, 14/21 next to 2x3. And we can keep doing this again and again. Each time we'll see the frequency for primes decrease and we'll see a new possibility for an even more composite number next to a prime opening up. Now we can see clearly that primes are always next to at least one composite number. As the numbers get larger and larger there's there's more possibility for the number next to a prime being more composite.