A360573 Odd numbers with exactly three zeros in their binary expansion.

17, 35, 37, 41, 49, 71, 75, 77, 83, 85, 89, 99, 101, 105, 113, 143, 151, 155, 157, 167, 171, 173, 179, 181, 185, 199, 203, 205, 211, 213, 217, 227, 229, 233, 241, 287, 303, 311, 315, 317, 335, 343, 347, 349, 359, 363, 365, 371, 373, 377, 399, 407, 411, 413

Offset

1,1

Comments

If m is a term then 2*m+1 is another term, since if M is the binary expansion of m, then M.1 where . stands for concatenation is the binary expansion of 2*m+1.

A052996 \ {1,3,8} is a subsequence, since for m >= 3, A052996(m) = 9*2^(m-2) - 1 has 100011..11 with m-2 trailing 1 for binary expansion.

A171389 \ {20} is a subsequence, since for m >= 1, A171389(m) = 21*2^m - 1 has 1010011..11 with m trailing 1 for binary expansion.

A198276 \ {18} is a subsequence, since for m >= 1, A198276(m) = 19*2^m - 1 has 1001011..11 with m trailing 1 for binary expansion.

Binary expansion of a(n) is A360574(n).

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

Numbers of the form 2^(a+1) - 2^b - 2^c - 2^d - 1 where a > b > c > d > 0. - Robert Israel, Feb 13 2023

Links

Table of n, a(n) for n=1..54.

Formula

A023416(a(n)) = 3.

Example

35_10 = 100011_2, so 35 is a term.

Maple

q:= n-> n::odd and add(1-i, i=Bits[Split](n))=3:
select(q, [$1..575])[];  # Alois P. Heinz, Feb 12 2023
# Alternative:
[seq(seq(seq(seq(2^(a+1) - 2^b - 2^c - 2^d - 1, d = c-1..1, -1), c=b-1..2, -1), b=a-1..3, -1), a=4..12)]; # Robert Israel, Feb 13 2023

Mathematica

Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 3 &] (* Amiram Eldar, Feb 12 2023 *)

Prog

(Python)
def ok(n): return n&1 and bin(n)[2:].count("0") == 3
print([k for k in range(414) if ok(k)]) # Michael S. Branicky, Feb 12 2023
(Python)
from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A360573_gen(): # generator of terms
    yield from (int('1'+''.join(d)+'1', 2) for l in count(0) for d in  multiset_permutations('000'+'1'*l))
A360573_list = list(islice(A360573_gen(), 54)) # Chai Wah Wu, Feb 18 2023
(Python)
from itertools import combinations, count, islice
def agen(): yield from ((1<<m)-(1<<i)-(1<<j)-(1<<k)-1 for m in count(5) for i, j, k in combinations(range(m-2, 0, -1), 3))
print(list(islice(agen(), 54))) # Michael S. Branicky, Feb 18 2023
(PARI) isok(m) = (m%2) && #select(x->(x==0), binary(m)) == 3; \\ Michel Marcus, Feb 13 2023

Crossrefs

Cf. A005408, A023416, A353654, A360574.

Subsequences: A052996 \ {1,3,8}, A171389 \ {20}, A198276 \ {18}.

Odd numbers with k zeros in their binary expansion: A000225 (k=0), A190620 (k=1), A357773 (k=2), this sequence (k=3).

Sequence in context: A033899 A110287 A102813 * A271536 A041568 A042305

Adjacent sequences: A360570 A360571 A360572 * A360574 A360575 A360576

Keyword

nonn,base

Author

Bernard Schott, Feb 12 2023

Status

approved