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

1,2

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

Links

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

Example

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.

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 *)

Prog

(PARI) upto(n) = {
    my(res = List());
    for(i = 0, logint(n, 2)+1,
        pow2 = 1<<i;
        listput(res, pow2);
            for(j = 0, i-2,
                listput(res, pow2 - 1<<j - 1);
            );
        );
        Set(res)}
(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<<d) - 1
        yield m
        yield from (b1-(m>>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

Crossrefs

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

Sequence in context: A353187 A389395 A099247 * A192583 A240064 A007192

Adjacent sequences: A384018 A384019 A384020 * A384022 A384023 A384024

Keyword

nonn,base,easy

Author

David A. Corneth, May 17 2025

Status

approved