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

%I A333535 #23 Oct 06 2024 09:16:24
%S A333535 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,
%T A333535 8,9,9,9,9,10,10,10,10,10,11,11,11,12,12,13,13,13,13,14,14,15,15,15,
%U A333535 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
%N A333535 Card{ k<=n, k such that all prime divisors of k are < sqrt(k) }.
%H A333535 Alois P. Heinz, <a href="/A333535/b333535.txt">Table of n, a(n) for n = 1..65536</a>
%p A333535 a:=[];
%p A333535 for n from 1 to 100 do
%p A333535    c:=0;
%p A333535    for m from 1 to n do
%p A333535       if A006530(m)^2 < m then c:=c+1; fi; od:
%p A333535 a:=[op(a),c];
%p A333535 od:
%p A333535 a;
%p A333535 # second Maple program:
%p A333535 a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+
%p A333535       `if`(max(map(i-> i[1], ifactors(n)[2]))^2<n, 1, 0))
%p A333535     end:
%p A333535 seq(a(n), n=1..100);  # _Alois P. Heinz_, Apr 10 2020
%t A333535 a[1] = 0;
%t A333535 a[n_] := a[n] = a[n-1] + Boole[Max[FactorInteger[n][[All, 1]]]^2 < n];
%t A333535 Array[a, 100] (* _Jean-François Alcover_, Nov 01 2020 *)
%o A333535 (Python)
%o A333535 from math import isqrt
%o A333535 from sympy import primepi
%o A333535 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
%Y A333535 The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.
%K A333535 nonn
%O A333535 1,12
%A A333535 _N. J. A. Sloane_, Apr 10 2020