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.
