|
%I #130 Feb 08 2024 01:38:38
%S 2,6,8,10,14,18,22,24,26,30,32,34,38,40,42,46,50,54,56,58,62,66,70,72,
%T 74,78,82,86,88,90,94,96,98,102,104,106,110,114,118,120,122,126,128,
%U 130,134,136,138,142,146,150,152,154,158,160,162,166,168,170,174
%N Numbers whose binary representation ends in an odd number of zeros.
%C Fraenkel (2010) called these the "dopey" numbers.
%C Also n such that A035263(n)=0 or A050292(n) = A050292(n-1).
%C Indices of even numbers in A033485. - _Philippe Deléham_, Mar 16 2004
%C 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
%C Indices of even numbers in A007913, in A001511. - _Philippe Deléham_, Mar 27 2004
%C This sequence consists of the increasing values of n such that A097357(n) is even. - _Creighton Dement_, Aug 14 2004
%C Numbers with an odd number of 2's in their prime factorization (e.g., 8 = 2*2*2). - _Mark Dow_, Sep 04 2007
%C Equals the set of natural numbers not in A003159 or A141290. - _Gary W. Adamson_, Jun 22 2008
%C 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. - _Gary W. Adamson_, Mar 20 2010
%C Refer to the comments in A003159 relating to A000041 and A174065. - _Gary W. Adamson_, Mar 21 2010
%C 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
%C a=(1,3,4,5,6,8,9,10,11,12,14,...) = A184427;
%C b=(2,7,12,21,31,44,58,74,...) = A184428.
%C Then putting s=a and repeating the operation gives
%C b'=(2,6,8,10,13,17,20,...), which has the same first four terms as A036554. - _Clark Kimberling_, Jan 14 2011
%C Or numbers having infinitary divisor 2, or the same, having factor 2 in Fermi-Dirac representation as a product of distinct terms of A050376. - _Vladimir Shevelev_, Mar 18 2013
%C Thus, numbers not in A300841 or in A302792. Equally, sequence 2*A300841(n) sorted into ascending order. - _Antti Karttunen_, Apr 23 2018
%H T. D. Noe, <a href="/A036554/b036554.txt">Table of n, a(n) for n = 1..1000</a>
%H L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-5/carlitz3-a.pdf">Representations for a special sequence</a>, Fib. Quart., 10 (1972), 499-518, 550 (see d(n) on page 501).
%H F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.
%H A. S. Fraenkel, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/">Home Page</a>
%H Aviezri S. Fraenkel, <a href="http://www.emis.de/journals/INTEGERS/papers/eg6/eg6.Abstract.html">New games related to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
%H Aviezri S. Fraenkel, <a href="http://dx.doi.org/10.1016/j.disc.2011.03.032">The vile, dopey, evil and odious game players</a>, Discrete Math. 312 (2012), no. 1, 42-46.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H Eric Sopena, <a href="http://arxiv.org/abs/1509.04199">i-Mark: A new subtraction division game</a>, arXiv:1509.04199 [cs.DM], 2015.
%H M. Stoll, <a href="http://arxiv.org/abs/1506.04286">Chabauty without the Mordell-Weil group</a>, arXiv preprint arXiv:1506.04286 [math.NT], 2015.
%H <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%F a(n) = A079523(n)+1 = A072939(n)-1.
%F a(n) = A003156(n) + n = A003157(n) - n = A003158(n) - n + 1. - _Philippe Deléham_, Apr 10 2004
%F Values of k such that A091297(k) = 2. - _Philippe Deléham_, Feb 25 2004
%F a(n) ~ 3n. - _Charles R Greathouse IV_, Nov 20 2012
%F a(n) = 2*A003159(n). - _Clark Kimberling_, Sep 30 2014
%F {a(n)} = A052330({A005408(n)}), where {a(n)} denotes the set of integers in the sequence. - _Peter Munn_, Aug 26 2019
%e From _Gary W. Adamson_, Mar 20 2010: (Start)
%e Equals terms in even numbered rows in the following multiplication table:
%e (rows are labeled 1,2,3,... as with the Towers of Hanoi disks)
%e 1, 3, 5, 7, 9, 11, ...
%e 2, 6, 10, 14, 18, 22, ...
%e 4, 12, 20, 28, 36, 44, ...
%e 8, 24, 40, 56, 72, 88, ...
%e ...
%e As shown, 2, 6, 8, 10, 14, ...; are in even numbered rows, given the row with (1, 3, 5, 7, ...) = row 1.
%e The term "5" is in an odd row, so the 5th Towers of Hanoi move is CW, moving disc #1 (in the first row).
%e a(3) = 8 in row 4, indicating that the 8th Tower of Hanoi move is CCW, moving disc #4.
%e A036554 bisects the positive nonzero natural numbers into those in the A036554 set comprising 1/3 of the total numbers, given sufficiently large n.
%e 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 is 1/3. (End)
%t Select[Range[200],OddQ[IntegerExponent[#,2]]&] (* _Harvey P. Dale_, Oct 19 2011 *)
%o (Haskell)
%o a036554 = (+ 1) . a079523 -- _Reinhard Zumkeller_, Mar 01 2012
%o (PARI) is(n)=valuation(n,2)%2 \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Magma) [2*m:m in [1..100] | Valuation(m,2) mod 2 eq 0]; // _Marius A. Burtea_, Aug 29 2019
%o (Python)
%o def ok(n):
%o c = 0
%o while n%2 == 0: n //= 2; c += 1
%o return c%2 == 1
%o print([m for m in range(1, 175) if ok(m)]) # _Michael S. Branicky_, Feb 06 2021
%o (Python)
%o from itertools import count, islice
%o def A036554_gen(startvalue=1): return filter(lambda n:(~n & n-1).bit_length()&1,count(max(startvalue,1))) # generator of terms >= startvalue
%o A036554_list = list(islice(A036554_gen(),30)) # _Chai Wah Wu_, Jul 05 2022
%Y Indices of odd numbers in A007814. Subsequence of A036552. Complement of A003159. Also double of A003159.
%Y Cf. A000041, A003157, A003158, A005408, A052330, A072939, A079523, A141290, A174065, A300841.
%K nonn,base,easy,nice
%O 1,1
%A _Tom Verhoeff_
%E Incorrect equation removed from formula by _Peter Munn_, Dec 04 2020
|
