Brain-explosion pattern of primes and the number 30?

Question

Prime numbers. Elusive little snips. They give you a warm trail with a dead end. Here's another one of those pattern 'trails': $$30$$ Normal number? How about 'expanding' outward? $$29, 30, 31$$ Yea, yea. Just twin primes. But if you go further... $$15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, >30<, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45$$ All the primes are rotating around 30. SPOILER: the pattern continues until you hit 11. Then you get 49. Bummer.

HOWEVER, there's better stuff coming.

Count up all the primes all the way up to 30. $$ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29$$ Now.

From 30 to 29 is 1
From 30 to 23 is 7.
From 30 to 19 is 11.
From 30 to 17 is 13.
From 30 to 13 is 17.
From 30 to 11 is 19 (yay even 11!)
From 30 to 7 is 23.
From 30 to 5 is...ARGH YOU SPOILED IT 25 YOU CURSED SEMIPRIME

Anyways from there is 27 and 28. He gave us the slip, boys.

I'm pretty sure, down deep in my bones, that there is another deeper trail hidden inside this one. If you doubt it, counting upward from 30 yields similar results... Also, why are these interrupting numbers (49, 25) all semiprimes? More about it later. Any comments?

EDIT: Goldbach's conjecture, perhaps, but how does that explain that adding UP from 30 yields the same prime number sequence?

Answer 1

The number $30$ is special in the sense that it is small, and that $30 = 2 \times 3 \times 5$ (the three smallest primes).

I think the key to your observation is the following "fact":

If $5 < n < 49$, $n$ is prime if and only if $\gcd(n,30) = 1$.

This is simply because $49 = 7^2$, so any nonprime number that is smaller must be divisible by $2$, $3$ or $5$.

Therefore, if $1 \leq k < 19$, $\gcd(30 + k,30) = 1 \Leftrightarrow \gcd(k,30) = 1 \Leftrightarrow \gcd(30 - k,30) = 1$, which explains why the primes appear to be "rotating around $30$". This fails for $k = 19$, when $30 + k = 49$.

For your second observation, it is again the fact that we have $\gcd(30-k,30) = 1 \Leftrightarrow \gcd(k,30) = 1$. When both $k$ and $30-k$ are greater than $5$, this means that both will be simultaneously prime. The observation fails exactly when $k = 25$ (hence $30-k = 5$).
Obs. This also explains why counting upward from $30$ exhibits the same pattern.