There are the Lucky Numbers https://en.wikipedia.org/wiki/Lucky_number https://en.wikipedia.org/wiki/Lucky_number. Generated by a variant of the Sieve of Eratos

There are the Lucky Numbers https://en.wikipedia.org/wiki/Lucky_number https://en.wikipedia.org/wiki/Lucky_number. Generated by a variant of the Sieve of Eratosthenes, they're believed to have a similar distribution to the primes while not having similar multiplicative properties.