A383667 Binary words beginning with 1 in which no binary digit occurs only once.

11, 111, 1001, 1010, 1100, 1111, 10001, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11100, 11111, 100001, 100010, 100011, 100100, 100101, 100110, 100111, 101000, 101001, 101010, 101011, 101100, 101101, 101110, 110000, 110001, 110010, 110011

Offset: 1

Author

Clark Kimberling, May 07 2025

Keywords

• nonn
• base

Crossrefs

Cf. A158581, A383666.

Programs

Mathematica

s = Select[Range[200], DigitCount[#, 2, 0] != 1 && DigitCount[#, 2, 1] != 1 &]
Map[First, RealDigits[s, 2]]; Map[FromDigits, m]

Python

def A383667(n):
    def f(x):
        if x<=1: return n+x
        l, s = x.bit_length(), bin(x)[2:]
        if (m:=s.count('0'))>0: return n+s.index('0')-(m>1)+(l*(l-1)>>1)
        return n-1+(l*(l+1)>>1)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return int(bin(m)[2:]) # Chai Wah Wu, May 21 2025