

A306998


List of low point records for pi(n)/n.


1



2, 9, 10, 16, 22, 25, 26, 27, 28, 35, 36, 40, 51, 52, 56, 57, 58, 66, 70, 78, 82, 86, 87, 88, 92, 93, 94, 95, 96, 121, 122, 123, 124, 125, 126, 135, 136, 145, 146, 147, 148, 162, 171, 172, 177, 178, 187, 188, 189, 190, 209, 210, 215, 216, 217, 218, 219, 220
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OFFSET

1,1


COMMENTS

Let f(n) = pi(n)/n, where pi(n) is the primecounting function (A000720). This sequence is the list of numbers n such that f(n) < f(k) for 2 <= k < n.
Because the primes generally become sparser forever, this list is infinite.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000


EXAMPLE

f(10) = 0.4, which is smaller than f(2), f(3), ... , f(9), so 10 is in the list.


MATHEMATICA

s={}; rm=1; Do[r = PrimePi[n]/n; If[r<rm, rm=r; AppendTo[s, n]], {n, 2, 1000}]; s (* Amiram Eldar, Mar 19 2019 *)


PROG

(Haskell)
 Very poor Haskell code, but let it stand until someone contributes the
 elegant threeliner that must exist. Its only merit is that it was actually
 used to produce the data given.
isPrime :: Int > Bool
isPrime = isPrime1 2
isPrime1 :: Int > Int > Bool
isPrime1 d n = n /= 1 && (d^2 > n  mod n d /= 0 && isPrime1 (d+1) n)
count :: (a > Bool) > [a] > Int
count f [] = 0
count f (x:xs) = (if f x then 1 else 0) + count f xs
pdf :: Int > Double
pdf n = fromIntegral (count isPrime [1..n]) / fromIntegral n
isRecord :: Int > Bool
isRecord n = (n == 2)  (pdf n) < (minimum (map pdf [2..(n1)]))
records :: [Int]
records = filter isRecord [2..100]


CROSSREFS

Cf. A000720.
Sequence in context: A051017 A078180 A058890 * A047468 A032929 A226832
Adjacent sequences: A306995 A306996 A306997 * A306999 A307000 A307001


KEYWORD

nonn,easy


AUTHOR

Allan C. Wechsler, Mar 18 2019


EXTENSIONS

More terms from Amiram Eldar, Mar 19 2019


STATUS

approved



