A190620 Odd numbers with a single zero in their binary expansion.

5, 11, 13, 23, 27, 29, 47, 55, 59, 61, 95, 111, 119, 123, 125, 191, 223, 239, 247, 251, 253, 383, 447, 479, 495, 503, 507, 509, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 3071, 3583, 3839, 3967, 4031

Offset: 1

Author

Reinhard Zumkeller, May 14 2011

Keywords

• nonn
• base

Comments

Odd numbers such that the binary weight is one less than the number of significant digits. Except for the initial 0, A129868 is a subsequence of this sequence. - Alonso del Arte, May 14 2011

From Bernard Schott, Oct 20 2022: (Start)

A036563 \ {-2, -1, 1} is a subsequence, since for m >= 3, A036563(m) = 2^m - 3 has 11..1101 with (m-2) starting 1's for binary expansion.

A083329 \ {1, 2} is a subsequence, since for m >= 2, A083329(m) = 3*2^(m-1) - 1 has 1011..11 with (m-1) trailing 1's for binary expansion.

A129868 \ {0} is a subsequence, since for m >= 1, A129868(m) = 2*4^m - 2^m - 1 is a binary cyclops number that has 11..11011..11 with m starting 1's and m trailing 1's for binary expansion.

The 0-bit position in binary expansion of a(n) is at rank A004736(n) + 1 from the right.

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

{2*a(n), n>0} form a subsequence of A353654 (numbers with one trailing 0 bit and one other 0 bit). (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007088, A023416, A190619, A353654.

A036563 \ {-2, -1, 1}, A083329 \ {1, 2}, A129868 are subsequences.

Odd numbers with k zeros in their binary expansion: A000225 (k=0), A357773 (k=2).

Programs

Haskell

import Data.List (elemIndices)
a190620 n = a190620_list !! (n-1)
a190620_list = filter odd $ elemIndices 1 a023416_list
-- A more efficient version, inspired by the Maple program in A190619:
a190620_list' = g 8 2 where
   g m 2 = (m - 3) : g (2*m) (m `div` 2)
   g m k = (m - k - 1) : g m (k `div` 2)

Maple

isA := proc(n) convert(n, base, 2): %[1] = nops(%) - add(%) end:
select(isA, [$1..4031]); # Peter Luschny, Oct 27 2022
# Alternatively, using a formula of Bernard Schott and A123578:
A190620 := proc(n) A123578(n); 4*2^% - 2^(1 - n + (% + %^2)/2) - 1 end:
seq(A190620(n), n = 1..50); # Peter Luschny, Oct 28 2022

Mathematica

Select[Range[1,5001,2],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Jul 12 2018 *)

Python

from itertools import count, islice
def agen():
    for d in count(3):
        b = 1 << d
        for i in range(2, d):
            yield b - (b >> i) - 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 13 2022

Python

from math import isqrt, comb
def A190620(n): return (1<<(a:=(isqrt(n<<3)+1>>1)+1)+1)-(1<Chai Wah Wu, Dec 18 2024

Formula

A190619(n) = A007088(a(n));

A023416(a(n)) = 1.

From Bernard Schott, Oct 21 2022: (Start)

a((n-1)*(n-2)/2 - (i-1)) = 2^n - 2^i - 1 for n >= 3 and 1 <= i <= n-2 (after Robert Israel in A357773).

a(n) = A000225(A002024(n)+2) - A000079(A004736(n)).

a(n) = 4*2^k(n) - 2^(1 - n + (k(n) + k(n)^2)/2) - 1, where k is the Kruskal-Macaulay function A123578.

A070939(a(n)) = A002024(n) + 2. (End)