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A366640 Lexicographically earliest sequence of distinct primes such that the sequence of squarefree numbers that are coprime to these primes has an asymptotic density 1/2.

%I A366640 #9 Oct 16 2023 09:30:18
%S A366640 5,79,1831,1856917,517136788981,25309896984298197131551,
%T A366640 9737146484866113825954170751740726870607451
%N A366640 Lexicographically earliest sequence of distinct primes such that the sequence of squarefree numbers that are coprime to these primes has an asymptotic density 1/2.
%C A366640 The corresponding sequence of squarefree numbers is A366641.
%C A366640 Equivalently, lexicographically earliest sequence of distinct primes such that Product_{n>=1} (1 + 1/a(n)) = 12/Pi^2.
%C A366640 The next term has 85 digits and is too large to be included in the data section.
%H A366640 Amiram Eldar, <a href="/A366640/b366640.txt">Table of n, a(n) for n = 1..11</a>
%e A366640 The asymptotic density of the squarefree numbers is 6/Pi^2 = 0.607... (A059956). Without the even numbers, the density of the odd squarefree numbers (A056911) is 4/Pi^2 = 0.405... (A185199), which is smaller than 1/2. Without the multiples of 3, the density of the squarefree numbers that are not divisible by 3 (A261034) is 9/(2*Pi^2) = 0.455... (A088245), which is also smaller than 1/2. Without the multiples of 5, the density of the squarefree numbers that are not divisible by 5 (A274546) is 5/Pi^2 = 0.506..., which is larger than 1/2. Therefore, a(1) = 5.
%e A366640 The asymptotic density of the squarefree numbers that are coprime to the primes a(1)..a(n), for n=1..8, is:
%e A366640   n  a(n)              density
%e A366640   -  ----------------  ------------------------------------------------------
%e A366640   1  5                 5/Pi^2 = 0.506605...
%e A366640   2  79                79/(16*Pi^2) = 0.500273...
%e A366640   3  1831              144649/(29312*Pi^2) = 0.500000269...
%e A366640   4  1856917           268601187133/(54429980416*Pi^2) = 0.500000000000966...
%e A366640   5  517136788981      1/2 + 1.975... * 10^(-23)
%e A366640   6  2.530... * 10^22  1/2 + 5.134... * 10^(-44)
%e A366640   7  9.737... * 10^42  1/2 + 3.775... * 10^(-85)
%e A366640   8  1.324... * 10^84  1/2 + 2.993... * 10^(-167)
%t A366640 seq[len_] := Module[{s = {}, r = 12/Pi^2, p}, Do[p = NextPrime[1/(r - 1)]; r *= (1/(1 + 1/p)); AppendTo[s, p], {len}]; s]; seq[8]
%Y A366640 Cf. A005117, A056911, A261034, A274546, A366641.
%Y A366640 Cf. A059956, A088245, A185199.
%K A366640 nonn
%O A366640 1,1
%A A366640 _Amiram Eldar_, Oct 15 2023