A230774 Number of primes less than first prime above square root of n.

1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Author

Jean-Christophe Hervé, Nov 01 2013

Keywords

• nonn
• easy

Comments

Or repeat k (prime(k)^2 - prime(k-1)^2) times, with prime(0) set to 0 for k = 1.

This sequence is useful to compute A055399 for prime numbers.

Examples

			a(5) = a(6) = a(7) = a(8) = a(9) = 2 because prime(1) = 2 < sqrt(5 to 9) <= prime(2) = 3.
		

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A050216, A056811, A055399, A230775.

Programs

Mathematica

Table[1 + PrimePi[Sqrt[n-1]], {n, 100}] (* Alonso del Arte, Nov 01 2013 *)

Python

from math import isqrt
from sympy import primepi
def A230774(n): return primepi(isqrt(n-1))+1 # Chai Wah Wu, Nov 04 2024

Formula

Repeat 1 prime(1)^2 = 4 times; for k>1, repeat k (prime(k)^2-prime(k-1)^2) = A050216(k-1) times.

a(n) - A056811(n) = characteristic function of squares of primes.