Prime numbers and "prim" numbers

If I understand your question (and I’m not sure I do), the answer is that the density of primes of the form 6n+1 and of the form 6n-1 are the same. About half the primes less than any large x are of the form 6n+1 and half of the form 6n-1. See here or here.

Only 7 is a prim number, because it’s leg is crossed. Or sitting up straight if you look from the side. All the others are obscene!

I mean, instead of counting the prime density in
(1,2,3,4,5,6,7,8,9,10,11,12,…),
count the density in
(5,7,11,13,17,19,23,25,29,31,…)

Since you’re looking at only a third of all numbers, do you modify the logarithmic interval three times as steep, or do you use some other function?

That’s easy: since the second list contains all the same primes as the first list (except for 2 and 3), but only a third as many numbers altogether (up to any particular x), the prime density of the second list is just like the prime density of the first, times three.

The fact I referred to is deeper and less obvious: that the primes in your second list are, in a sense, equally distributed between {5, 11, 17, 23, 29, 35, …} and {7, 13, 19, 25, 37, …}.

Just wanted to say thank for such a lucid (several in the thread but yours in particular) explanation.