

A036554


Numbers whose binary representation ends in an odd number of zeros.


66



2, 6, 8, 10, 14, 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 50, 54, 56, 58, 62, 66, 70, 72, 74, 78, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 110, 114, 118, 120, 122, 126, 128, 130, 134, 136, 138, 142, 146, 150, 152, 154, 158, 160, 162, 166, 168, 170, 174
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OFFSET

1,1


COMMENTS

Fraenkel (2010) called these the "dopey" numbers.
Also n such that A035263(n)=0 or A050292(n) = A050292(n1).
Indices of even numbers in A033485.  Philippe Deléham, Mar 16 2004
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 even numbers in A007913, in A001511.  Philippe Deléham, Mar 27 2004
This sequence consists of the increasing values of n such that A097357(n) is even.  Creighton Dement, Aug 14 2004
Numbers with an odd number of 2's in their prime factorization (e.g., 8 = 2*2*2).  Mark Dow, Sep 04 2007
Equals the set of natural numbers not in A003159 or A141290.  Gary W. Adamson, Jun 22 2008
Represents the set of CCW nth moves in the standard Tower of Hanoi game; and terms in even rows of a [1, 3, 5, 7, 9, ...] * [1, 2, 4, 8, 16, ...] multiplication table. Refer to the example.  Gary W. Adamson, Mar 20 2010
Refer to the comments in A003159 relating to A000041 and A174065.  Gary W. Adamson, Mar 21 2010
If the upper sWythoff sequence of s is s, then s=A036554. (See A184117 for the definition of lower and upper sWythoff sequences.) Starting with any nondecreasing sequence s of positive integers, A036554 is the limit when the upper 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
b'=(2,6,8,10,13,17,20,...), which has the same first four terms as A036554.  Clark Kimberling, Jan 14 2011
Or numbers having infinitary divisor 2, or the same, having factor 2 in FermiDirac representation as a product of distinct terms of A050376.  Vladimir Shevelev, Mar 18 2013
Thus, numbers not in A300841 or in A302792. Equally, sequence 2*A300841(n) sorted into ascending order.  Antti Karttunen, Apr 23 2018


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499518, 550 (see d(n) on page 501).
A. S. Fraenkel, Home Page
Aviezri S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 4246.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Eric Sopena, iMark: A new subtraction division game, arXiv:1509.04199 [cs.DM], 2015.
M. Stoll, Chabauty without the MordellWeil group, arXiv preprint arXiv:1506.04286 [math.NT], 2015.
Index entries for 2automatic sequences.


FORMULA

a(n) = A079523(n)+1 = A072939(n)1 = A056196(n)/4.
a(n) = A003158(n)  n + 1 = A003157(n)  n.  Philippe Deléham, Feb 22 2004
Values of k such that A091297(k) = 2.  Philippe Deléham, Feb 25 2004
a(n) = A003156(n) + n.  Philippe Deléham, Apr 10 2004
a(n) ~ 3n.  Charles R Greathouse IV, Nov 20 2012
a(n) = 2*A003159(n)  Clark Kimberling, Sep 30 2014
{a(n)} = A052330({A005408(n)}), where {a(n)} denotes the set of integers in the sequence.  Peter Munn, Aug 26 2019


EXAMPLE

From Gary W. Adamson, Mar 20 2010: (Start)
Equals terms in even numbered rows in the following multiplication table:
(rows are labeled 1,2,3,... as with the Towers of Hanoi disks)
1, 3, 5, 7, 9, 11, ...
2, 6, 10, 14, 18, 22, ...
4, 12, 20, 28, 36, 44, ...
8, 24, 40, 56, 72, 88, ...
...
As shown, 2, 6, 8, 10, 14, ...; are in even numbered rows, given the row with (1, 3, 5, 7, ...) = row 1.
The term "5" is in an odd row, so the 5th Towers of Hanoi move is CW, moving disc #1 (in the first row).
a(3) = 8 in row 4, indicating that the 8th Tower of Hanoi move is CCW, moving disc #4.
A036554 bisects the positive nonzero natural numbers into those in the A036554 set comprising 1/3 of the total numbers, given sufficiently large n.
This corresponds to 1/3 of the TOH moves being CCW and 2/3 CW. Row 1 of the multiplication table = 1/2 of the natural numbers, row 2 = 1/4, row 3 = 1/8 and so on, or 1 = (1/2 + 1/4 + 1/8 + 1/16 + ...). Taking the oddindexed terms of this series given offset 1, we obtain 2/3 = 1/2 + 1/8 + 1/32 + ..., while sum of the evenindexed terms is 1/3. (End)


MAPLE

op(select(n>(searchtext(`1`, convert(convert(n, base, 2), string))2)/3 mod 2=1, [$1..174])); # Paolo P. Lava, Oct 30 2018


MATHEMATICA

f[n_Integer] := Block[{k = n, c = 0}, While[ EvenQ[k], c++; k /= 2]; c]; Select[ Range[185], OddQ[ f[ # ]] & ] (* Or *)
a[n_] := a[n] = If[n < 2, 1, n  a[Floor[n/2]]]; t = Table[a[n], {n, 1, 300}]; Union[ Select[t, t[[ # ]] == t[[ #  1]] & ]]
Select[Range[200], OddQ[IntegerExponent[#, 2]]&] (* Harvey P. Dale, Oct 19 2011 *)


PROG

(Haskell)
a036554 = (+ 1) . a079523  Reinhard Zumkeller, Mar 01 2012
(PARI) is(n)=valuation(n, 2)%2 \\ Charles R Greathouse IV, Nov 20 2012
(MAGMA) [2*m:m in [1..100]Valuation(m, 2) mod 2 eq 0]; // Marius A. Burtea, Aug 29 2019


CROSSREFS

Indices of odd numbers in A007814. Subsequence of A036552. Complement of A003159. Also double of A003159.
Cf. A000041, A003157, A003158, A005408, A052330, A056196, A072939, A079523, A141290, A174065, A300841.
Sequence in context: A047395 A284794 A187692 * A260400 A296387 A302658
Adjacent sequences: A036551 A036552 A036553 * A036555 A036556 A036557


KEYWORD

nonn,base,easy,nice


AUTHOR

Tom Verhoeff


STATUS

approved



