A071520 Number of 5-smooth numbers (A051037) <= n.

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 27, 27, 28, 28

Offset: 1

Author

Benoit Cloitre, Jun 02 2002

Keywords

• easy
• nonn

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008683, A051037, A106598, A112751.

Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), this sequence (p=5), A071604 (p=7), A071523 (p=11), A080684 (p=13), A080685 (p=17), A080686 (p=19).

Programs

Mathematica

Accumulate[Table[If[Max[FactorInteger[n][[;;,1]]]<6,1,0],{n,80}]] (* Harvey P. Dale, Aug 04 2024 *)

PARI

for(n=1,100,print1(sum(k=1,n,if(sum(i=4,n,if(k%prime(i),0,1)),0,1)),","))

PARI

a(n)=-sum(k=1,n,moebius(2*3*5*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007

Python

from sympy import integer_log
def A071520(n):
    c = 0
    for i in range(integer_log(n,5)[0]+1):
        for j in range(integer_log(m:=n//5**i,3)[0]+1):
            c += (m//3**j).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

Formula

a(n) = Card{ k | A051037(k) <= n }.

Asymptotically : let a = 1/(6*log(2)*log(3)*log(5)) and b = sqrt(30) then a(n) = a*log(b*n)^3 + O(log(n)).

a(n) = -Sum_{k=1,n} mu(30*k)*floor(n/k). - Benoit Cloitre, Jun 14 2007

a(n) = Sum_{i=0..floor(log_5(n))} Sum_{j=0..floor(log_3(n/5^i))} floor(log_2(2*n/(5^i*3^j))). - Ridouane Oudra, Jul 17 2020

Extensions

Title corrected by Rainer Rosenthal, Aug 30 2020