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A339344
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Lexicographically earliest sequence of odd primes such that the asymptotic density of the numbers which are divisible by at least one of these primes is 1/2.
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2
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OFFSET
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1,1
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COMMENTS
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Given a set of prime numbers P, finite or infinite, the set of numbers which are divisible by at least one of the primes in P has an asymptotic density Product_{p in P} (1 - 1/p). If P is finite, then this density is equal to 1/2 only when P = {2}. Otherwise, the density is 1/2 for infinitely many sets P. This sequence is the lexicographically earliest infinite sequence of such primes.
The first 5 terms are the Fermat primes (A019434).
a(10) = 7.455916... * 10^135 is too large to be included in the data section.
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LINKS
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FORMULA
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a(1) = 3, a(n) = nextprime(r(n-1)/(r(n-1) - 1/2)), where r(n) = Product_{k=1..n-1} 1 - 1/a(n).
Product_{n=>1} (1 - 1/a(n)) = 1/2.
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MATHEMATICA
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s = {}; r = 1; p = 3; Do[AppendTo[s, p]; r *= 1 - 1/p; p = NextPrime[r/(r - 1/2)], {9}]; s
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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