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A036554
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Numbers n such that their binary representation ends in odd number of zeros.
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36
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Fraenkel (2010) called these the "dopey" numbers.
Also n such that A035263(n)=0 or A050292(n)==A050292(n-1).
Indices of even numbers in A033485. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 16 2004
Indices of even numbers in A007913, in A001511. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 27 2004
This sequence consists of the increasing values of n such that A097357(n) is even. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 14 2004
Numbers with an odd number of 2's in their prime factorization (e.g. 8 = 2*2*2). - Mark Dow (dow(AT)uoregon.edu), Sep 04 2007
Equals the set of natural numbers not in A003159 or A141290. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 22 2008
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 20 2010: (Start)
Represents the set of CCW n-th 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. (End)
Refer to the comments in A003159 relating to A000041 and A174065. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2010]
If the upper s-Wythoff sequence of s is s, then s=A036554. (See A184117 for the definition of lower and upper s-Wythoff sequences.) Starting with any nondecreasing sequence s of positive integers, A036554 is the limit when the upper s-Wythoff operation is iterated. For example, starting with s=(1,4,9,16,...)=(n^2), we obtain lower and upper s-Wythoff 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. [From Clark Kimberling, ck6(AT)evansville.edu Jan 14 2011]
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REFERENCES
| L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550 (see d(n) on page 501).
Aviezri S. Fraenkel, The vile, dopy, evil and odious game players, Discrete Math., to appear (2010).
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
A. S. Fraenkel, Home Page
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
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FORMULA
| a(n) = A003158(n) - n + 1 = A003157(n) - n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 22 2004
Values of k such that A091297(k) = 2 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 25 2004
a(n) = A003156(n) + n. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Apr 10 2004
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EXAMPLE
| Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 20 2010: (Start)
Equals terms in even numbered rows in the following multiplication table:
(rows are labeled 1,2,3,...as with the Tower of Hanoi discs.)
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 5-th Tower of Hanoi move is CW, moving
disc #1 (in the first row).
a(3) = 8 in row 4, indicating that the 8-th Tower of Hanoi move is CCW, moving
disc #4.
A036554 bisects the positive non-zero 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 odd indexed terms of
this series given offset 1, we obtain 2/3 = 1/2 + 1/8 + 1/32 + ...; while
sum of the even indexed terms = 1/3. (End)
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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]]&] (* From Harvey P. Dale, Oct 19 2011 *)
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CROSSREFS
| Indices of odd numbers in A007814. Subsequence of A036552. Complement of A003159. Also double of A003159.
Cf. A003157 A003158 A003159.
a(n) = A079523(n)+1 = A072939(n)-1 = A056196(n)/4.
Cf. A141290.
Cf. A000041, A174065 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2010]
Sequence in context: A141105 A047395 A187692 * A084909 A038619 A118257
Adjacent sequences: A036551 A036552 A036553 * A036555 A036556 A036557
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Tom Verhoeff (Tom.Verhoeff(AT)acm.org)
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EXTENSIONS
| More terms from Mark Dow (dow(AT)uoregon.edu), Sep 04 2007
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