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Here is a sequence something like the "hailstone" one : starting from any number, if it is prime, multiply by two and add one, if composite, replace by its largest prime factor. This depends on there being a lot of primes of the form 2p+1 where P is prime. I don't know enough math to know how long those go on, but there are interesting sequencers for up to the 1000th prime.

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