A383666 -id:A383666 - cp's OEIS frontend

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

Clark Kimberling, May 07 2025

Cf. A158581, A383666.

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

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

A384021 Powers of 2 along with numbers one power of 2 less than binary repunits, but the power of two subtracted does not flip the leading bit.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 11, 13, 14, 16, 23, 27, 29, 30, 32, 47, 55, 59, 61, 62, 64, 95, 111, 119, 123, 125, 126, 128, 191, 223, 239, 247, 251, 253, 254, 256, 383, 447, 479, 495, 503, 507, 509, 510, 512, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1024, 1535, 1791, 1919

Offset: 1

David A. Corneth, May 17 2025

Also numbers where in the binary expansion the bit 0 or 1 occurs exactly once.

Union of A000079 and A164874. - Chai Wah Wu, May 21 2025

			4 = 2^2 is term as it is a power of 2.
5 is a term as 5 = (2^3 - 1) - 2^1; a power of two less than a binary repunit and subtracting 2 from 7 does not flip the most significant bit of 7.

David A. Corneth, Table of n, a(n) for n = 1..10011

Cf. A383666 (complement), A000079, A030130, A164874.

A384021list[k_] := Flatten[{1, Table[{2^i - 1 - BitShiftRight[2^i, Range[2, i]], 2^i}, {i, 2 - Boole[k == 1], k}]}]; (* returns terms up to 2^k *)
A384021list[11] (* Paolo Xausa, Jun 12 2025 *)

upto(n) = {
    my(res = List());
    for(i = 0, logint(n, 2)+1,
        pow2 = 1<

from itertools import count, islice
def agen(): # generator of terms
    yield from (1, 2)
    for d in count(3):
        m, b1 = 1<<(d-1), (1<>i) for i in range(1, d))
print(list(islice(agen(), 58))) # Michael S. Branicky, May 18 2025

def A384021(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):
        if x<=1: return n
        l, s = x.bit_length(), bin(x)[2:]
        if (m:=s.count('0'))>0: return n+x-s.index('0')+(m>1)-(l*(l-1)>>1)
        return n+x+1-(l*(l+1)>>1)
    return bisection(f,n,n) # Chai Wah Wu, May 21 2025

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

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A384021 Powers of 2 along with numbers one power of 2 less than binary repunits, but the power of two subtracted does not flip the leading bit.

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Mathematica

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Python

Python