A384210 Number of numbers <= n of the form p * m^2, where p is a prime and m is an integer >= 1.

0, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 13, 13, 13, 14, 15, 16, 16, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 28, 28, 28, 28, 29, 29, 30, 30, 31, 31, 31, 31, 32, 33, 33, 33, 34, 35, 36, 36, 37

Offset: 1

Author

Ridouane Oudra, May 22 2025

Keywords

• nonn

Comments

Partial sums of A358769.

First differs from A055038 at a(30).

a(A229125(n)) = n.

Links

• Ridouane Oudra, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A229125, A358769, A000720, A008836, A001221.

Programs

Maple

with(numtheory): A358769:=n-> add(nops(factorset(d)), d in divisors(n)) mod 2:
seq(add(A358769(i), i=1..n), n=1..100);

PARI

a(n) = sum(k=1, n, isprime(core(k))); \\ Michel Marcus, May 29 2025

Python

from math import isqrt
from sympy import primepi
def A384210(n): return sum(primepi(n//y**2) for y in range(1,isqrt(n)+1)) # Chai Wah Wu, Jun 06 2025

Formula

a(n) = Sum_{i=1..n} A358769(i).

a(n) = Sum_{i=1..floor(sqrt(n))} pi(floor(n/i^2)), where pi = A000720.

a(n) = - Sum_{i=1..n} lambda(i)*omega(i)*floor(n/i), where lambda = A008836 and omega = A001221.