

A059396


Number of primes less than square root of nth prime; i.e., number of trial divisions by smaller primes to show that nth 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



