A357774 Binary expansions of odd numbers with two zeros in their binary expansion.

1001, 10011, 10101, 11001, 100111, 101011, 101101, 110011, 110101, 111001, 1001111, 1010111, 1011011, 1011101, 1100111, 1101011, 1101101, 1110011, 1110101, 1111001, 10011111, 10101111, 10110111, 10111011, 10111101, 11001111, 11010111, 11011011, 11011101, 11100111, 11101011

Offset: 1

Author

Bernard Schott, Oct 19 2022

Keywords

• nonn
• base

Comments

For m >= 4, there are A000217(m-3) terms with m digits.

Crossrefs

Cf. A000217, A007088, A357773.

A267524 \ {1, 10, 100} and A267705 \ {1, 10} are two subsequences.

Similar, but with k zeros in their binary expansion: A000042 (k=0), A190619 (k=1).

Programs

Mathematica

FromDigits[IntegerDigits[#, 2]] & /@ Select[Range[1, 250, 2], DigitCount[#, 2, 0] == 2 &] (* Amiram Eldar, Oct 19 2022 *)

PARI

isok(k) = (k%2) && (#binary(k) == hammingweight(k)+2); \\ A357773
f(n) = fromdigits(binary(n), 10); \\ A007088
lista(nn) = apply(f, select(isok, [1..nn])); \\ Michel Marcus, Oct 19 2022

Python

from itertools import combinations, count, islice
def agen(): # generator of terms
    for d in count(4):
        b, c = 2**d - 1, 2**(d-1)
        for i, j in combinations(range(1, d-1), 2):
            yield int(bin(b - (c >> i) - (c >> j))[2:])
print(list(islice(agen(), 30))) # Michael S. Branicky, Oct 19 2022

Python

from itertools import count, islice
def A357774_gen(): # generator of terms
    for l in count(2):
        m = (10**(l+2)-1)//9
        for i in range(l,0,-1):
            k = m-10**i
            yield from (k-10**j for j in range(i-1,0,-1))
A357774_list = list(islice(A357774_gen(),30)) # Chai Wah Wu, Feb 19 2023

Python

from math import isqrt, comb
from sympy import integer_nthroot
def A357774(n):
    a = (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2,3))+3
    b = isqrt((j:=comb(a-1,3)-n+1)<<3)+3>>1
    c = j-comb((r:=isqrt(w:=j<<1))+(w>r*(r+1)),2)
    return (10**a-1)//9-10**b-10**c # Chai Wah Wu, Dec 19 2024

Formula

a(n) = A007088(A357773(n)).