A080193 5-smooth numbers which are not 3-smooth.

5, 10, 15, 20, 25, 30, 40, 45, 50, 60, 75, 80, 90, 100, 120, 125, 135, 150, 160, 180, 200, 225, 240, 250, 270, 300, 320, 360, 375, 400, 405, 450, 480, 500, 540, 600, 625, 640, 675, 720, 750, 800, 810, 900, 960, 1000, 1080, 1125, 1200, 1215, 1250, 1280, 1350

Offset: 1

Author

Klaus Brockhaus, Feb 10 2003

Keywords

• easy
• nonn

Comments

Numbers of the form 2^r*3^s*5^t with r, s >= 0, t > 0.

That is, 5-smooth numbers which are multiples of 5. - Charles R Greathouse IV, Mar 19 2015

Examples

			15 = 3*5 is a term but 18 = 2*3^2 is not.
		

Links

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

Crossrefs

Cf. A051037, A003586.

Programs

Mathematica

Select[Range[1000], FactorInteger[#][[-1, 1]] == 5 &] (* Amiram Eldar, Nov 10 2020 *)

PARI

{m=1440; z=[]; for(r=0,floor(log(m)/log(2)),a=2^r; for(s=0,floor(log(m/a)/log(3)),b=a*3^s; for(t=1, floor(log(m/b)/log(5)),z=concat(z,b*5^t)))); z=vecsort(z); for(i=1,length(z),print1(z[i],","))}

PARI

list(lim)=my(v=List(),x=1,y,z); while((x*=5)<=lim, y=x/3; while((y*=3)<=lim, z=y/2; while((z*=2)<=lim, listput(v, z)))); Set(v) \\ Charles R Greathouse IV, Mar 19 2015

Python

from sympy import integer_log
def A080193(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,5)[0]+1):
            for j in range(integer_log(y:=x//5**i,3)[0]+1):
                c -= (y//3**j).bit_length()
        return c
    return bisection(f,n,n)*5 # Chai Wah Wu, Sep 16 2024

Python

# faster for initial segment of sequence
import heapq
from itertools import islice
def A080193gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 5*v
            oldv = v
            for p in psmooth_primes:
                    heapq.heappush(h, v*p)
print(list(islice(A080193gen(), 55))) # Michael S. Branicky, Sep 18 2024

Formula

From Amiram Eldar, Nov 10 2020: (Start)

a(n) = 5 * A051037(n).

Sum_{n>=1} 1/a(n) = 3/4. (End)