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%I #34 Jul 11 2019 14:44:06
%S 1,2,3,5,7,8,9,11,12,13,15,17,19,20,21,23,25,27,28,29,31,32,33,35,36,
%T 37,39,41,43,44,45,47,48,49,51,52,53,55,57,59,60,61,63,65,67,68,69,71,
%U 73,75,76,77,79,80,81,83,84,85,87,89,91,92,93,95,97,99,100,101,103,105
%N For n > 1: if n is present, 2n is not.
%C The Name line gives a property of the sequence, not a definition. The sequence can be defined simultaneously with b(n) := A171945(n) via a(n) = mex{a(i), b(i) : 0 <= i < n} (n >= 0}, b(n)=2a(n). The two sequences are complementary, hence A053661 is identical to A171944 (except for the first terms). Furthmore, A053661 is the same as A003159 except for the replacement of vile by dopey powers of 2. - _Aviezri S. Fraenkel_, Apr 28 2011
%C For n >= 2, either n = 2^k where k is odd or n = 2^k*m where m > 1 is odd and k is even (found by Kirk Bresniker and Stan Wagon). [_Robert Israel_, Oct 10 2010]
%C Subsequence of A175880; A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.
%H Robert Israel, <a href="/A053661/b053661.txt">Table of n, a(n) for n = 1..10000</a>
%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.
%p N:= 1000: # to get all terms <= N
%p sort([1,seq(2^(2*i+1),i=0..(ilog2(N)-1)/2), seq(seq(2^(2*i)*(2*j+1),j=1..(N/2^(2*i)-1)/2),i=0..ilog2(N)/2)]); # _Robert Israel_, Jul 24 2015
%t Clear[T]; nn = 105; T[n_, k_] := T[n, k] = If[n < 1 || k < 1, 0, If[n == 1 || k == 1, 1, If[k > n, T[k, n], If[n > k, T[k, Mod[n, k, 1]], -Product[T[n, i], {i, n - 1}]]]]]; DeleteCases[Table[If[T[n, n] == -1, n, ""], {n, 1, nn}], ""] (* _Mats Granvik_, Aug 25 2012 *)
%o (Haskell)
%o a053661 n = a053661_list !! (n-1)
%o a053661_list = filter (> 0) a175880_list -- _Reinhard Zumkeller_, Feb 09 2011
%Y Essentially identical to A171944 and the complement of A171945.
%K nonn,easy
%O 1,2
%A Jeevan Chana Rai (Karanjit.Rai(AT)btinternet.com), Feb 16 2000
%E More terms from _James A. Sellers_, Feb 22 2000
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