

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



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 *)
DeleteDuplicates[Table[{n, PrimePi[n]/n}, {n, 2, 250}], LessEqual[#1[[2]], #2[[2]]]&][[;; , 1]] (* Harvey P. Dale, May 30 2023 *)


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



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



