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.

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

Author

David A. Corneth, May 17 2025

Keywords

• nonn
• base
• easy

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Python

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

Python

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