A085126 Multiples of 3 which are members of A002473. Or multiples of 3 with the largest prime divisor < 10.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 36, 42, 45, 48, 54, 60, 63, 72, 75, 81, 84, 90, 96, 105, 108, 120, 126, 135, 144, 147, 150, 162, 168, 180, 189, 192, 210, 216, 225, 240, 243, 252, 270, 288, 294, 300, 315, 324, 336, 360, 375, 378, 384, 405, 420, 432, 441, 450

Offset: 1

Author

Amarnath Murthy, Jul 06 2003

Keywords

• easy
• nonn

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10001 (first 1001 terms from Harvey P. Dale)

Crossrefs

Intersection of A008585 and A002473.

Cf. A085125, A085127, A085128, A085129, A080194, A085131, A085132.

Programs

Mathematica

Select[3*Range[200],FactorInteger[#][[-1,1]]<10&] (* Harvey P. Dale, Apr 10 2019 *)

Python

from sympy import integer_log
def A085126(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,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    return bisection(f,n,n)*3 # Chai Wah Wu, Sep 17 2024

Python

# faster for initial segment of sequence
import heapq
from itertools import islice
def A085126gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5, 7]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 3*v
            oldv = v
            for p in psmooth_primes:
                    heapq.heappush(h, v*p)
print(list(islice(A085126gen(), 65))) # Michael S. Branicky, Sep 17 2024

Formula

a(n) = 3*A002473(n). - Chai Wah Wu, Sep 18 2024

Sum_{n>=1} 1/a(n) = 35/24. - Amiram Eldar, Sep 23 2024

Extensions

More terms from David Wasserman, Jan 28 2005

Offset changed to 1 by Michael S. Branicky, Sep 17 2024