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Some hint:
https://protokol2020.wordpress.com/2017/03/07/landaus-problem-continuation/
Not a useful hint, sorry. "About half" of primes are in that sequence. The product of (1-2/p) behaves like exp of the sum of -2/p, which looks like a constant times log log N. So your product diverges to 0.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should start on Monday, and end on Sunday.
4. Unflag the two options "Notify me of new top level comments on this article" and "