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

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, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 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

Offset

2,7

Comments

Number of k <= n such that the greatest prime factor of k is <= log(n).

Links

Alois P. Heinz, Table of n, a(n) for n = 2..65536

Formula

a(n) = A096300(n), n>2. - R. J. Mathar, Apr 27 2020

Maple

A333534 := n -> nops(select(k -> A006530(k) <= ilog(n), [$1..n])):
seq(A333534(n), n=2..86); # Peter Luschny, Apr 09 2020
# second Maple program:
b:= proc(n) option remember; max(1, map(i-> i[1], ifactors(n)[2])) end:
a:= n-> (t-> add(`if`(b(i)<= t, 1, 0), i=1..n))(ilog(n)):
seq(a(n), n=2..100);  # Alois P. Heinz, Apr 09 2020

Mathematica

a[n_] := Select[Range[n], FactorInteger[#][[-1, 1]] <= Log[n]&] // Length;
a /@ Range[2, 100] (* Jean-François Alcover, May 17 2020 *)

Prog

(PARI) gpf(j)={if(j==1, 1, my(f=factor(j)); f[#f[, 2], 1])};
for(n=2, 80, my(L=log(n)); print1(sum(k=1, n, gpf(k)<=L), ", ")) \\ Hugo Pfoertner, Apr 09 2020
(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
a(n)=if(n<8, return(n>2)); sm(n, precprime(log(n))) \\ Charles R Greathouse IV, Apr 16 2020

Crossrefs

Cf. A001671, A006530, A051102, A137845, A295084.

Sequence in context: A066014 A080678 A096300 * A372512 A035672 A350716

Adjacent sequences: A333531 A333532 A333533 * A333535 A333536 A333537

Keyword

nonn

Author

N. J. A. Sloane, Apr 08 2020

Status

approved