A085125 Even numbers which are 7-smooth.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 50, 54, 56, 60, 64, 70, 72, 80, 84, 90, 96, 98, 100, 108, 112, 120, 126, 128, 140, 144, 150, 160, 162, 168, 180, 192, 196, 200, 210, 216, 224, 240, 250, 252, 256, 270, 280, 288, 294, 300, 320, 324, 336

Offset: 1

Author

Amarnath Murthy, Jul 06 2003

Keywords

• nonn
• easy

Comments

Equivalently, multiples of 2 with the largest prime divisor < 10.

Crossrefs

Intersection of A005843 and A002473.

Cf. A085126, A085127, A085128, A085129, A080194, A085131, A085132.

Programs

Mathematica

Select[2*Range[200],FactorInteger[#][[-1,1]]<10&] (* Harvey P. Dale, Jul 06 2018 *)

Python

from sympy import integer_log
def A085125(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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()-1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 31 2025

Formula

From Amiram Eldar, Sep 23 2024: (Start)

a(n) = 2*A002473(n).

Sum_{n>=1} 1/a(n) = 35/16. (End)

Extensions

More terms from David Wasserman, Jan 28 2005

Offset changed by Andrew Howroyd, Sep 19 2024