The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 2
5, 79, 1831, 1856917, 517136788981, 25309896984298197131551, 9737146484866113825954170751740726870607451 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The corresponding sequence of squarefree numbers is A366641.
Equivalently, lexicographically earliest sequence of distinct primes such that Product_{n>=1} (1 + 1/a(n)) = 12/Pi^2.
The next term has 85 digits and is too large to be included in the data section.
LINKS
EXAMPLE
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)
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]
CROSSREFS
Sequence in context: A198152 A197232 A152297 * A244585 A293786 A141828
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 15 2023
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 24 00:44 EDT 2024. Contains 372765 sequences. (Running on oeis4.)