

A079523


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


33



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
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OFFSET

1,2


COMMENTS

Also, n such that A010060(n) = A010060(n+1) where A010060 is the ThueMorse 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 ThueMorse sequence, arXiv preprint arXiv:1401.3727 [math.NT], 2014.
J.P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the ThueMorse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375388.
J.P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, Simon Plouffe, B. E. Sagan, A relative of the ThueMorse sequence, Discrete Math., 139, 1995, 455461.
Narad Rampersad, Manon Stipulanti, The Formal Inverse of the PeriodDoubling Sequence, arXiv:1807.11899 [math.CO], 2018.
Thomas Zaslavsky, AntiFibonacci Numbers: A Formula, Sep 26 2016 [Introduces the name "utterly odd".  N. J. A. Sloane, Sep 27 2016]
Index entries for 2automatic sequences.


FORMULA

a(n) is asymptotic to 3n.
a(n) = 2*A003159(n)  1. a(1)=1, a(n) = a(n1) + 2 if (a(n1)+1)/2 does not belong to the sequence and a(n)=a(n1)+4 otherwise.  Emeric Deutsch and Bruce E. Sagan, Apr 02 2003
a(n) = (1/2)A081706(2n1).
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


MATHEMATICA

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


PROG

(Haskell)
import Data.List (elemIndices)
a079523 n = a079523_list !! (n1)
a079523_list = elemIndices 0 a035263_list
 Reinhard Zumkeller, Mar 01 2012
(PARI) is(n)=valuation(n+1, 2)%2 \\ Charles R Greathouse IV, Mar 07 2013
(MAGMA) [n: n in [0..200]  Valuation(n+1, 2) mod 2 eq 0 + 1]; // Vincenzo Librandi, Apr 16 2015


CROSSREFS

Cf. A003159, A003157, A003158, A075326, A131323.
Sequence in context: A284742 A111083 A050550 * A231271 A039504 A097280
Adjacent sequences: A079520 A079521 A079522 * A079524 A079525 A079526


KEYWORD

nonn,base,easy


AUTHOR

Benoit Cloitre, Jan 21 2003


STATUS

approved



