A079523 Utterly odd numbers: numbers whose binary representation ends in an odd number of ones.

1, 5, 7, 9, 13, 17, 21, 23, 25, 29, 31, 33, 37, 39, 41, 45, 49, 53, 55, 57, 61, 65, 69, 71, 73, 77, 81, 85, 87, 89, 93, 95, 97, 101, 103, 105, 109, 113, 117, 119, 121, 125, 127, 129, 133, 135, 137, 141, 145, 149, 151, 153, 157, 159, 161, 165, 167, 169, 173, 177, 181

Offset: 1

Author

Benoit Cloitre, Jan 21 2003

Keywords

• nonn
• base
• easy

Comments

Also, n such that A010060(n) = A010060(n+1) where A010060 is the Thue-Morse sequence.

Sequence of n such that a(n) = 3n begins 7, 23, 27, 29, 31, 39, 71, 87, 91, 93, 95, ...

Values of k such that the Motzkin number A001006(2k) is even. Values of k such that the number of restricted hexagonal polyominoes with 2k+1 cells is even (see A002212). Values of k such that the number of directed animals of size k+1 is even (see A005773). Values of k such that the Riordan number A005043(k) is even. - Emeric Deutsch and Bruce E. Sagan, Apr 02 2003

a(n) = A036554(n)-1 = A072939(n)-2. - Ralf Stephan, Jun 09 2003

Odious and evil terms alternate. - Vladimir Shevelev, Jun 22 2009

The sequence has the following fractal property: remove terms of the form 4k+1 from the sequence, and the remaining terms are of the form 4k+3: 7, 23, 31, 39, 55, 71, 87, ...; then subtract 3 from each of these terms and divide by 4 and you get the original sequence: 1, 5, 7, 9, 13, ... - Benoit Cloitre, Apr 06 2010

A035263(a(n)) = 0. - Reinhard Zumkeller, Mar 01 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv preprint arXiv:1401.3727 [math.NT], 2014.
• J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
• J.-P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, Simon Plouffe, and B. E. Sagan, A relative of the Thue-Morse sequence, Discrete Math., 139, 1995, 455-461.
• Narad Rampersad and Manon Stipulanti, The Formal Inverse of the Period-Doubling Sequence, arXiv:1807.11899 [math.CO], 2018.
• Thomas Zaslavsky, Anti-Fibonacci Numbers: A Formula, Sep 26 2016 [Introduces the name "utterly odd". - _N. J. A. Sloane_, Sep 27 2016]
• Index entries for 2-automatic sequences.

Crossrefs

Cf. A003159, A003157, A003158, A075326, A131323.

Programs

Haskell

import Data.List (elemIndices)
a079523 n = a079523_list !! (n-1)
a079523_list = elemIndices 0 a035263_list
-- Reinhard Zumkeller, Mar 01 2012

Magma

[n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 + 1]; // Vincenzo Librandi, Apr 16 2015

Mathematica

Select[ Range[200], MatchQ[ IntegerDigits[#, 2], {b : (1) ..} | {_, 0, b : (1) ..} /; OddQ[ Length[{b}]]] & ] (* Jean-François Alcover, Jun 17 2013 *)

PARI

is(n)=valuation(n+1,2)%2 \\ Charles R Greathouse IV, Mar 07 2013

Python

from itertools import count, islice
def A079523_gen(startvalue=1): return filter(lambda n:(~(n+1)&n).bit_length()&1,count(max(startvalue,1))) # generator of terms >= startvalue
A079523_list = list(islice(A079523_gen(),30)) # Chai Wah Wu, Jul 05 2022

Python

def A079523(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):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n)-1 # Chai Wah Wu, Jan 29 2025

Formula

a(n) is asymptotic to 3n.

a(n) = 2*A003159(n) - 1. a(1)=1, a(n) = a(n-1) + 2 if (a(n-1)+1)/2 does not belong to the sequence and a(n) = a(n-1) + 4 otherwise. - Emeric Deutsch and Bruce E. Sagan, Apr 02 2003

a(n) = (1/2)*A081706(2n-1).

a(n) = A003158(n) - n = A003157(n) - n - 1. - Philippe Deléham, Feb 22 2004

Values of k such that A091297(k) = 0. - Philippe Deléham, Feb 25 2004