A357773 Odd numbers with two zeros in their binary expansion.

9, 19, 21, 25, 39, 43, 45, 51, 53, 57, 79, 87, 91, 93, 103, 107, 109, 115, 117, 121, 159, 175, 183, 187, 189, 207, 215, 219, 221, 231, 235, 237, 243, 245, 249, 319, 351, 367, 375, 379, 381, 415, 431, 439, 443, 445, 463, 471, 475, 477, 487, 491, 493, 499, 501

Offset

1,1

Comments

A048490 \ {1} is a subsequence, since for m >= 1, A048490(m) = 8*2^m - 7 has 11..11001 with m starting 1 for binary expansion.

A153894 \ {4} is a subsequence, since for m >= 1, A153894(m) = 5*2^m - 1 has 10011..11 with m trailing 1 for binary expansion.

A220236 is a subsequence, since for m >= 1, A220236(m) = 2^(2*m + 2) - 2^(m + 1) - 2^m - 1 has 11..110011..11 with m starting 1 and m trailing 1 for binary expansion.

For k > 2, there are (k-1)*(k-2)/2 terms between 2^k and 2^(k+1), or equivalently (k-1)*(k-2)/2 terms with k+1 bits.

Binary expansion of a(n) is A357774(n).

{4*a(n), n>0} form a subsequence of A353654 (numbers with two trailing 0 bits and two other 0 bits).

Links

Table of n, a(n) for n=1..55.

Formula

A023416(a(n)) = 2.

a((n-1)*(n-2)*(n-3)/6 - (i-1)*(i-2)/2 - (j-1)) = 2^n - 2^i - 2^j - 1 for 1 <= j < i <= n-2. - Robert Israel, Oct 13 2022

Maple

seq(seq(seq(2^n-1-2^i-2^j, j=i-1..1, -1), i=n-2..1, -1), n=4..10); # Robert Israel, Oct 13 2022

Mathematica

Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 2 &] (* Amiram Eldar, Oct 12 2022 *)

Prog

(Python)
def a(n):
    m = 0
    while m*(m+1)*(m+2)//6 <= n: m += 1
    m -= 1 # m = A056556(n-1)
    k, r, j = m + 4, n - m*(m+1)*(m+2)//6, 0
    while r >= 0: r -= (m+1-j); j += 1
    j += 1
    return 2**k - 2**(k-j) - 2**(-r) - 1
print([a(n) for n in range(60)]) # Michael S. Branicky, Oct 12 2022
(Python) # faster version for generating initial segment of sequence
from itertools import combinations, count, islice
def agen():
    for d in count(4):
        b, c = 2**d - 1, 2**(d-1)
        for i, j in combinations(range(1, d-1), 2):
            yield b - (c >> i) - (c >> j)
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 13 2022
(PARI) isok(k) = (k%2) && (#binary(k) == hammingweight(k)+2); \\ Michel Marcus, Oct 13 2022
(PARI) list(lim)=my(v=List()); for(n=4, logint(lim\=1, 2)+1, my(N=2^n-1); forstep(a=n-2, 2, -1, my(A=N-1<<a); forstep(b=a-1, 1, -1, my(t=A-1<<b); if(t>lim, break(2)); listput(v, t)))); Vec(v) \\ Charles R Greathouse IV, Oct 21 2022

Crossrefs

Cf. A018900, A005408, A023416, A353654, A357774.

Odd numbers with k zeros in their binary expansion: A000225 (k=0), A190620 (k=1).

Subsequences: A048490 \ {1}, A153894 \ {4}, A220236.

Sequence in context: A157140 A244538 A242332 * A075981 A079368 A167529

Adjacent sequences: A357770 A357771 A357772 * A357774 A357775 A357776

Keyword

nonn,base,easy

Author

Bernard Schott, Oct 12 2022

Extensions

a(11) and beyond from Michael S. Branicky, Oct 12 2022

Status

approved