By introducing modular objects associated to the sequel of rings $$(Z/2,Z/6,Z/30,Z/210,Z/2310,..., Z/p_n\#Z)$$ a sequence of coefficients is updated$$(2;\color{green}{\frac83}; 3.2;...$$ (see my answer to my proper question here)
I've been running computers and I've observed that
- There are twice more primes $p$ such that $p+6$ is also prime (sexys) than twins;
- $\color{green}{\frac83}$ more primes $p$ such that $p+30$ is also prime than twins;
- $3.2$ more $p$ such that $p+210$ is also prime than twins;
- ...
Until now, I had always had the greatest difficulty in deciphering how to find these constants in Wolphram Alpha's article about k-Tuples conjectures.
For example, for $p+$30, where I'm looking for $\color{green}{\frac83}$ , I understand that I am interested in what the article notes $$\boxed{C(15)=2\prod_{q\text{ prime}}^{}\frac{q(q-2)}{(q-1)^2}\prod_{q|15}^{}\frac{q-1}{q-2}}$$
In fact, the initial purpose of my post was even to ask for help to do so but I think I'm starting to understand since $$\prod_{q|15}^{}\frac{q-1}{q-2}=\frac{3-1}{3-2}\frac{5-1}{5-2}=2\times\frac{4}{3}=\color{green}{\frac{8}{3}}$$
1.- I still have to think about it a little more, maybe get your help but it will do it.
2.- But, suddenly, a more interesting question arises for me: is this approach in modular arithmetic to highlight the Hardy and Littlewood coefficients, but especially to give them a non-statistical modular meaning (see here) , known?
Definitions, notations and examples:
- $p_1=2, p_2=3, p_3=5, p_4=7, p_5=11, ... , $ are the primes;
- $\forall n\in \mathbb N^*, p_n\#:=p_1\times p_2\times ....\times p_n$;
- examples : $p_1\#=2, p_2\#=p_1\times p_2=2\times 3=6, p_3\#=2.3.5=30, ...$
