What do we know about Lucky numbers?

Question

I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics.

Wolfram states: write "out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number >1 is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, .... Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ...."

They have their very own Goldbach, Legendre, Lemoine and twin conjectures. I was wondering whether there have been some significant discoveries about this numbers. One question which really intrigues me is the existence of a Riemann-like function which, when applied similarly to Riemann Zeta function, it gives you the exact number of lucky numbers less than n. I'm pretty sure we don't even know it exists, but I think it would be nice if it did. Also would be very cool if we could know the process behind the creation of prime-like sequences such as this one or practical numbers.

Answer 1

There seems to be a lot of open questions about the lucky numbers, so let me phrase what we know about them and what we don't. Most of the questions are written as the equivalent forms of some known results for the prime numbers.

What we know

• The Lucky Numbers Theorem (equivalent form of the Prime Number Theorem for lucky numbers) - if we let $\pi_{s}(x)$ be the number of lucky numbers $\leqslant x$, then $$\lim_{x \to \infty}\frac{\pi_{s}(x) \log x}{x} = 1.$$ This result can actually be deduced from the asymptotic formula for the $n-$th lucky number $s_n$, given by S. Chowla here, $$ s_n = n\log n + \frac{n}{2}(\log \log n)^2 + o(n (\log \log n)^2).$$

What we don't know

• Infinitude of the lucky prime numbers - are there infinitely many numbers that are both lucky and prime?

• Twin lucky numbers - are there infinitely many numbers $s$ such that both $s$ and $s+2$ are lucky numbers?

• The Goldbach conjecture for lucky numbers - does for any integer $n \geqslant 1$ the equation $$2n = s_1 + s_2$$ has a solution for $s_1, s_2$ being lucky?

• Dirichlet's theorem for the lucky numbers - if we let $\pi_s(x;q,a)$ be the number of lucky numbers $s \leqslant x$, $s \equiv a (\mod{q})$, then how does $\pi_{s}(x;q,a)$ behave as $x$ tend to infinity?

Most of the problems associated with the lucky numbers are due to the fact that sieving process consists of iterations done not always on the whole set of all positive integers, but on the set given after the previous iteration. This makes everything a lot more complicated, as the Riemann zeta function and zeta-like functions are mostly suitable for the sets defined by iterations of sieving processes, all done on the same set.

In my little paper in polish (here) I have proven that, if we let $A_{k}$ be the set of all the integers sifted in the $k-$th iteration of the process (so every $s_k-$th of the remaining), then there exists a positive integer $q_k$ and some set $R_k \subset \{0,1,\ldots,q_{k}-1\}$ such that $$ A_{k} = \{iq_k + r \colon i \geqslant 0, r \in R_{k}\}$$ where $i \geqslant 0$ runs through nonnegative integers. I was looking for a theorem of that type as it could allow to use tools like the Dirichlet's theorem or related in a preferable form (like Brun-Titchmarsch inequality or Siegel-Walfisz theorem) to solve the problem of lucky primes. However, due to some complications in the sieving process at places where $q_k > x$, it still isn't enough. However, it does indeed show that with some (infinitely many, in fact) choices of $q$ and $a$ we'll have $\pi_{s}(x;q,a)=0$ for all $x>0$.