A059396 Number of primes less than square root of n-th prime; i.e., number of trial divisions by smaller primes to show that n-th prime is indeed prime.

0, 0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9

Offset

1,5

Comments

Asymptotic to 2*(n/log(n))^(1/2):

Since p_n ~ n * log n, a(n) ~ sqrt(n * log n) / (log (sqrt(n * log n))) ~ 2 * sqrt(n) * sqrt(log n) / (log n + log log n) ~ 2 * sqrt(n / log n). - Daniel Forgues, Sep 04 2018

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000720(A000196(A000040(n))).

Example

a(32) = 5 since the 32nd prime is 131 which is not divisible by 2, 3, 5, 7 or 11 (and does not need to be tested against 13, 17, 19 etc. since 13^2 = 169 > 131).

Maple

a:= proc(n) option remember;
      numtheory[pi](floor(sqrt(ithprime(n))))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 05 2011

Mathematica

Table[PrimePi[Sqrt[Prime[n]]], {n, 110}] (* Harvey P. Dale, Sep 06 2015 *)

Prog

(PARI) a(n) = primepi(sqrtint(prime(n))); \\ Altug Alkan, Sep 05 2018

Crossrefs

Sequence in context: A000196 A111850 A343265 * A108602 A305025 A363743

Adjacent sequences: A059393 A059394 A059395 * A059397 A059398 A059399

Keyword

nonn,easy

Author

Henry Bottomley, Jan 29 2001

Status

approved