A032937 Numbers k whose base-2 representation Sum_{i=0..m} d(i)*2^(m-i) has d(i)=0 for all odd i, excluding 0. Here m is the position of the leading bit of k.

1, 2, 4, 5, 8, 10, 16, 17, 20, 21, 32, 34, 40, 42, 64, 65, 68, 69, 80, 81, 84, 85, 128, 130, 136, 138, 160, 162, 168, 170, 256, 257, 260, 261, 272, 273, 276, 277, 320, 321, 324, 325, 336, 337, 340, 341, 512, 514, 520, 522, 544, 546

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

Essentially the same as A126684. - R. J. Mathar, Jun 15 2008

A126684 is the primary entry for this sequence. - Franklin T. Adams-Watters, Aug 30 2014

Crossrefs

Cf. A033053, A126684.

Programs

Mathematica

Join[{1},Select[Range[0,600],Union[Take[IntegerDigits[#,2],{2,-1,2}]]=={0}&]] (* Harvey P. Dale, Sep 17 2023 *)

Python

from gmpy2 import digits
def A032937(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 g(x):
        s = digits(x,4)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return int(s,2)
        return int(s[:i]+'1'*(l-i),2)
    def f(x): return n+x-g(x)-g(x>>1)
    return bisection(f,n,n) # Chai Wah Wu, Oct 29 2024