A339344 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.

3, 5, 17, 257, 65537, 4294967311, 1229782942255939601, 88962710886098567818446141338419231, 255302062200114858892457591448999891874349780170241684791167583265041

Offset: 1

Author

Amiram Eldar, Nov 30 2020

Keywords

• nonn

Comments

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..12

Crossrefs

Cf. A000215, A019434, A080307, A125045, A262228, A339345.

Programs

Mathematica

s = {}; r = 1; p = 3; Do[AppendTo[s, p]; r *= 1 - 1/p; p = NextPrime[r/(r - 1/2)], {9}]; s

Formula

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.