

A003159


Numbers n whose binary representation ends in an even number of zeros.
(Formerly M2306)


50



1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101, 103, 105
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OFFSET

1,2


COMMENTS

Fraenkel (2010) called these the "vile" numbers.
Minimal with respect to property that parity of number of 1's in binary expansion alternates.
Minimal with respect to property that sequence is half its complement. [Corrected by Aviezri S. Fraenkel, Jan 29 2010]
If n appears then 2n does not.
Increasing sequence of positive integers n such that A035263(n)=1 (from paper by Allouche et al.).  Emeric Deutsch, Jan 15 2003
a(n) is an odious number (see A000069) for n odd; a(n) is an evil number (see A001969) for n even.  Philippe Deléham, Mar 16 2004
Indices of odd numbers in A007913, in A001511.  Philippe Deléham, Mar 27 2004
Partial sums of A026465.  Paul Barry, Dec 09 2004
A121701(2*a(n)) = A121701(a(n)); A096268(a(n)1) = 0.  Reinhard Zumkeller, Aug 16 2006
A different permutation of the same terms may be found in A141290 and the accompanying array.  Gary W. Adamson, Jun 14 2008
a(n) = nth clockwise Tower of Hanoi move; counterclockwise if not in the sequence.  Gary W. Adamson, Jun 14 2008
Indices of terms of ThueMorse sequence A010060 which are different from the previous term.  Tanya Khovanova, Jan 06 2009
The sequence has the following fractal property. Remove the odd numbers from the sequence, leaving 4,12,16,20,28,36,44,48,52,... Dividing these terms by 4 we get 1,3,4,5,7,9,11,12,..., which is the original sequence back again.  Benoit Cloitre, Apr 06 2010
From Gary W. Adamson, Mar 21 2010: (Start)
A conjectured identity relating to the partition sequence, A000041 as polcoeff p(x); A003159, and its characteristic function A035263: (1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1,...); and A036554 indicating nth terms with zeros in A035263: (2, 6, 8, 10, 14, 18, 22,...).
The conjecture states that p(x) = A(x) = A(x^2) when A(x) = polcoeff A174065 = the Euler transform of A035263 = 1/(1x)*(1x^3)*(1x^4)*(1x^5)* ...=(1 + x + x^2 + 2x^3 + 3x^4 + 4x^5 + ...) and the aerated variant = the Euler transform of the complement of A035263: 1/(1x^2)*(1x^6)*(1x^8)* ...=(1 + x^2 + x^4 + 2x^6 + 3x^8 + 4x^10 + ...).
(End)
The conjecture above was proved by JeanPaul Allouche on Dec 21 2013.  Gary W. Adamson, Jan 22 2014
If the lower sWythoff sequence of s is s, then s=A003159. (See A184117 for the definition of lower and upper sWythoff sequences.) Starting with any nondecreasing sequence s of positive integers, A003159 is the limit when the lower sWythoff operation is iterated. For example, starting with s=(1,4,9,16,...)=(n^2), we obtain lower and upper sWythoff sequences
a=(1,3,4,5,6,8,9,10,11,12,14...)=A184427;
b=(2,7,12,21,31,44,58,74,...)=A184428.
Then putting s=a and repeating the operation gives a'=(1,3,4,5,7,9,11,12,14,...), which has the same first eight terms as A003159.  Clark Kimberling, Jan 14 2011


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Lars Blomberg and T. D. Noe, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe)
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, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of ThueMorse, Discrete Math., 139 (1995), 455461.
J.P. Allouche and Jeffrey Shallit, The Ubiquitous ProuhetThueMorse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, SpringerVerlag, 1999, pp. 116.
J.P. Allouche, J. Shallit and G. Skordev, Selfgenerating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 115.
George E. Andrews and David Newman, Binary Representations and Theta Functions, 2017.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499518, 550.
R. Clerico, P. Fabbri e F. Ortenzio, Pericolosamente vicino a Collatz, Rudi Mathematici, N. 226 (Nov 2017), p. 14 (in Italian).
M. Domaratzki, Trajectorybased codes, Acta Informatica, Volume 40, Numbers 67 / May, 2004.
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191215.
A. Dubickas, On the distance from a rational power to the nearest integer, Journal of Number Theory, Volume 117, Issue 1, March 2006, Pages 222239.
A. Dubickas, On a sequence related to that of ThueMorse and its applications, Discrete Math. 307 (2007), no. 910, 10821093. MR2292537 (2008b:11086).
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
A. S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 4246.
Aviezri S. Fraenkel, Home Page
C. Kimberling, Problem E2850, Amer. Math. Monthly, 87 (1980), 671.
C. Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
C. Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147160.
E. Sopena, iMark: A new subtraction division game, arXiv:1509.04199 [cs.DM], 2015.
D. Wakeham and D. R. Wood, On multiplicative Sidon sets, INTEGERS, 13 (2013), #A26.
Index entries for 2automatic sequences.
Index entries for sequences related to binary expansion of n


FORMULA

a(0)=1; for n>=0, a(n+1) = a(n)+1 if (a(n)+1)/2 is not already in the sequence, = a(n)+2 otherwise.
a(n) = A056196(n)/8.
Lim n >infinity a(n)/n = 3/2  Benoit Cloitre, Jun 13 2002
More precisely, a(n) = 3n/2 + O(log n).  Charles R Greathouse IV, Sep 23 2012
a(n) = Sum_{k=1..n} A026465(k))  Benoit Cloitre, May 31 2003
a(n+1) = (if a(n) mod 4 = 3 then A007814(a(n) + 1) mod 2 else a(n) mod 2) + a(n) + 1; a(1) = 1.  Reinhard Zumkeller, Aug 03 2003
a(A003157(n)) is even.  Philippe Deléham, Feb 22 2004
Sequence consists of numbers of the form 4^i*(2j+1), i>=0, j>=0.  Jon Perry, Jun 06 2004
G.f.: (1/(1x))Product{k>=1, 1+x^A001045(k)}.  Paul Barry, Dec 09 2004
a(1)=1, a(2)=3 and for n>=2 we get a(n+1)=4+a(n)+a(n1)a(a(n)n+1)a(a(n1)n+2)  Benoit Cloitre, Apr 08 2010
If A(x) is the counting function for a(n)<=x, then A(2^n)=(2^(n+1)+(1)^n)/3.  Vladimir Shevelev, Apr 15 2010
a(n) = A121539(n) + 1.  Reinhard Zumkeller, Mar 01 2012
A003159 = { N  A007814(N) is even }.  M. F. Hasler, Oct 29 2013


EXAMPLE

1=1, 3=11, 5=101 and 7=111 have no (0 = even) trailing zeros, 4=100 has 2 (= even) trailing zeros in the base2 representation.
2=10 and 6=110 end in one (=odd number) of trailing zeros in their base2 representation, therefore are not members of this sequence.  M. F. Hasler, Oct 29 2013


MAPLE

filter:= n > type(padic:ordp(n, 2), even):
select(filter, [$1..1000]); # Robert Israel, Jul 07 2014


MATHEMATICA

f[n_Integer] := Block[{k = n, c = 0}, While[ EvenQ[k], c++; k /= 2]; c]; Select[ Range[105], EvenQ[ f[ # ]] & ]
Select[Range[150], EvenQ[IntegerExponent[#, 2]]&] (* Harvey P. Dale, Oct 19 2011 *)


PROG

(PARI) a(n)=if(n<2, n>0, n=a(n1); until(valuation(n, 2)%2==0, n++); n)
(PARI) is(n)=valuation(n, 2)%2==0 \\ Charles R Greathouse IV, Sep 23 2012
(Haskell)
import Data.List (delete)
a003159 n = a003159_list !! (n1)
a003159_list = f [1..] where f (x:xs) = x : f (delete (2*x) xs)
 Reinhard Zumkeller, Nov 04 2011


CROSSREFS

For the actual binary numbers see A280049.
Indices of even numbers in A007814.
Complement of A036554, also onehalf of A036554.
Cf. A001285, A010060, A056196, A000041, A174065, A292118.
Sequence in context: A224985 A282808 A260401 * A187691 A141259 A047501
Adjacent sequences: A003156 A003157 A003158 * A003160 A003161 A003162


KEYWORD

nonn,nice,easy,eigen,base


AUTHOR

N. J. A. Sloane, Simon Plouffe


EXTENSIONS

Additional comments from Michael Somos
Edited by M. F. Hasler, Oct 29 2013


STATUS

approved



