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A036554
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Numbers whose binary representation ends in an odd number of zeros.
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96
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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
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OFFSET
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1,1
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COMMENTS
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Fraenkel (2010) called these the "dopey" numbers.
Numbers with an odd number of 2's in their prime factorization (e.g., 8 = 2*2*2). - Mark Dow, Sep 04 2007
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
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
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
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LINKS
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FORMULA
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EXAMPLE
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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 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)
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MATHEMATICA
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Select[Range[200], OddQ[IntegerExponent[#, 2]]&] (* Harvey P. Dale, Oct 19 2011 *)
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PROG
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(Haskell)
(Magma) [2*m:m in [1..100] | Valuation(m, 2) mod 2 eq 0]; // Marius A. Burtea, Aug 29 2019
(Python)
def ok(n):
c = 0
while n%2 == 0: n //= 2; c += 1
return c%2 == 1
(Python)
from itertools import count, islice
def A036554_gen(startvalue=1): return filter(lambda n:(~n & n-1).bit_length()&1, count(max(startvalue, 1))) # generator of terms >= startvalue
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CROSSREFS
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KEYWORD
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nonn,base,easy,nice
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AUTHOR
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EXTENSIONS
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Incorrect equation removed from formula by Peter Munn, Dec 04 2020
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STATUS
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approved
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