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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A178786 A000196 A111850 * A108602 A305025 A085290

Adjacent sequences:  A059393 A059394 A059395 * A059397 A059398 A059399

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jan 29 2001

STATUS

approved

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Last modified November 20 08:16 EST 2018. Contains 317385 sequences. (Running on oeis4.)