A096300 -id:A096300 - cp's OEIS frontend

A071604 a(n) is the number of 7-smooth numbers <= n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 34, 35, 36, 36, 36, 36, 36, 36, 37, 37, 38

Offset: 1

Benoit Cloitre, Jun 02 2002

A 7-smooth number is a number of the form 2^x*3^y*5^z*7^u, (x,y,z,u) >= 0.

In other words, a 7-smooth number is a number with no prime factor greater than 7. - Peter Munn, Nov 20 2021

			a(11) = 10 as there are 10 7-smooth numbers <= 11. Namely 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - _David A. Corneth_, Apr 19 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Partial sums of A086299.
Column 7 of A080786.
Cf. A002473, A106600, A343598.
Equivalent sequences with other limits on greatest prime factor: A070939 (2), A071521 (3), A071520 (5), A071523 (11), A080684 (13), A080685 (17), A080686 (19), A096300 (log n).

for(n=1,100,print1(sum(k=1,n,if(sum(i=5,n,if(k%prime(i),0,1)),0,1)),","))

from sympy import integer_log
def A071604(n):
    c = 0
    for i in range(integer_log(n,7)[0]+1):
        i7 = 7**i
        m = n//i7
        for j in range(integer_log(m,5)[0]+1):
            j5 = 5**j
            r = m//j5
            for k in range(integer_log(r,3)[0]+1):
                c += (r//3**k).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = Card{ k | A002473 (k) <= n }.

Name corrected by David A. Corneth, Apr 19 2021

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

Original entry on oeis.org

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

N. J. A. Sloane, Apr 08 2020

nonn

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

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

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

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

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

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

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

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

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A071604 a(n) is the number of 7-smooth numbers <= n.

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

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PARI

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