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A333534 a(n) is the number of log(n)-smooth numbers <= n.

%I A333534 #36 May 17 2020 07:29:16
%S A333534 0,1,1,1,1,1,4,4,4,4,4,4,4,4,5,5,5,5,5,10,10,10,11,11,11,12,12,12,12,
%T A333534 12,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,
%U A333534 16,16,16,16,16,16,16,16,16,16,17,17,17,17,17,17,17,17,18,18,18,18,18,18,18,18,18,19,19,19,19,19,19
%N A333534 a(n) is the number of log(n)-smooth numbers <= n.
%C A333534 Number of k <= n such that the greatest prime factor of k is <= log(n).
%H A333534 Alois P. Heinz, <a href="/A333534/b333534.txt">Table of n, a(n) for n = 2..65536</a>
%F A333534 a(n) = A096300(n), n>2. - _R. J. Mathar_, Apr 27 2020
%p A333534 A333534 := n -> nops(select(k -> A006530(k) <= ilog(n), [$1..n])):
%p A333534 seq(A333534(n), n=2..86); # _Peter Luschny_, Apr 09 2020
%p A333534 # second Maple program:
%p A333534 b:= proc(n) option remember; max(1, map(i-> i[1], ifactors(n)[2])) end:
%p A333534 a:= n-> (t-> add(`if`(b(i)<= t, 1, 0), i=1..n))(ilog(n)):
%p A333534 seq(a(n), n=2..100);  # _Alois P. Heinz_, Apr 09 2020
%t A333534 a[n_] := Select[Range[n], FactorInteger[#][[-1, 1]] <= Log[n]&] // Length;
%t A333534 a /@ Range[2, 100] (* _Jean-François Alcover_, May 17 2020 *)
%o A333534 (PARI) gpf(j)={if(j==1,1,my(f=factor(j));f[#f[,2],1])};
%o A333534 for(n=2,80,my(L=log(n));print1(sum(k=1,n,gpf(k)<=L),", ")) \\ _Hugo Pfoertner_, Apr 09 2020
%o A333534 (PARI) sm(lim, p)=if(p==2, return(logint(lim\1, 2)+1)); my(s=0, q=precprime(p-1), t=1); for(e=0, logint(lim\=1, p), s+=sm(lim\t, q); t*=p); s
%o A333534 a(n)=if(n<8,return(n>2)); sm(n, precprime(log(n))) \\ _Charles R Greathouse IV_, Apr 16 2020
%Y A333534 Cf. A001671, A006530, A051102, A137845, A295084.
%K A333534 nonn
%O A333534 2,7
%A A333534 _N. J. A. Sloane_, Apr 08 2020