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

1,2

Comments

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

Links

Table of n, a(n) for n=1..73.

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A051037, A106598, A112751.

Sequence in context: A287354 A025548 A121604 * A195918 A371156 A176842

Adjacent sequences: A071517 A071518 A071519 * A071521 A071522 A071523

Keyword

easy,nonn

Author

Benoit Cloitre, Jun 02 2002

Extensions

Title corrected by Rainer Rosenthal, Aug 30 2020

Status

approved