The lucky numbers are defined by a sieve, which results in numbers that asymptotically mirror the prime density $\sim n / \log n$: $$ 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, \ldots $$ OEIS A000959.
The Green-Tao Theorem established that the primes contain arbitrarily long arithmetic progressions.
Q. Do the lucky numbers contain arbitrarily long arithmetic progressions?