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.
5, 79, 1831, 1856917, 517136788981, 25309896984298197131551, 9737146484866113825954170751740726870607451
Offset: 1
Keywords
Examples
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. The asymptotic density of the squarefree numbers that are coprime to the primes a(1)..a(n), for n=1..8, is: n a(n) density - ---------------- ------------------------------------------------------ 1 5 5/Pi^2 = 0.506605... 2 79 79/(16*Pi^2) = 0.500273... 3 1831 144649/(29312*Pi^2) = 0.500000269... 4 1856917 268601187133/(54429980416*Pi^2) = 0.500000000000966... 5 517136788981 1/2 + 1.975... * 10^(-23) 6 2.530... * 10^22 1/2 + 5.134... * 10^(-44) 7 9.737... * 10^42 1/2 + 3.775... * 10^(-85) 8 1.324... * 10^84 1/2 + 2.993... * 10^(-167)
Links
- Amiram Eldar, Table of n, a(n) for n = 1..11
Programs
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Mathematica
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]
Comments