A333535 Card{ k<=n, k such that all prime divisors of k are < sqrt(k) }.

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 21, 21, 22, 23, 23, 23, 24, 24, 24, 24, 24, 24, 25

Offset: 1

Author

N. J. A. Sloane, Apr 10 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..65536

Crossrefs

The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.

Programs

Maple

a:=[];
for n from 1 to 100 do
   c:=0;
   for m from 1 to n do
      if A006530(m)^2 < m then c:=c+1; fi; od:
a:=[op(a),c];
od:
a;
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+
      `if`(max(map(i-> i[1], ifactors(n)[2]))^2Alois P. Heinz, Apr 10 2020

Mathematica

a[1] = 0;
a[n_] := a[n] = a[n-1] + Boole[Max[FactorInteger[n][[All, 1]]]^2 < n];
Array[a, 100] (* Jean-François Alcover, Nov 01 2020 *)

Python

from math import isqrt
from sympy import primepi
def A333535(n): return int(n-1-primepi(n//(m:=isqrt(n)))+sum(primepi(i)-primepi(n//i) for i in range(1,m))) # Chai Wah Wu, Oct 06 2024