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A092246 Odd "odious" numbers (A000069).

%I A092246 #41 Mar 03 2023 17:14:36
%S A092246 1,7,11,13,19,21,25,31,35,37,41,47,49,55,59,61,67,69,73,79,81,87,91,
%T A092246 93,97,103,107,109,115,117,121,127,131,133,137,143,145,151,155,157,
%U A092246 161,167,171,173,179,181,185,191,193,199,203,205,211,213,217,223,227,229,233
%N A092246 Odd "odious" numbers (A000069).
%C A092246 In other words, numbers having a binary representation ending in 1, and an odd number of 1's overall. It follows that by decrementing an odd odious number, one gets an even evil number (A125592). - _Ralf Stephan_, Aug 27 2013
%C A092246 The members of the sequence may be called primitive odious numbers because every odious number is a power of 2 times one of these numbers. Note that the difference between consecutive terms is either 2, 4, or 6. - _T. D. Noe_, Jun 06 2007
%C A092246 From _Gary W. Adamson_, Apr 06 2010: (Start)
%C A092246 a(n) = A026147(n)-th odd number, where A026147 = (1, 4, 6, 7, 10, 11, ...); e.g.,
%C A092246          n: 1   2   3   4   5   6   7   8   9  10  11
%C A092246   n-th odd: 1   3   5   7   9  11  13  15  17  19  21
%C A092246       a(n): 1           7      11  13          19  21
%C A092246 etc. (End)
%C A092246 Numbers m, such that when merge-sorting lists of length m, the maximal number of comparisons is even: A003071(a(n)) = A230720(n). - _Reinhard Zumkeller_, Oct 28 2013
%C A092246 Fixed points of permutation pair A268717/A268718. - _Antti Karttunen_, Feb 29 2016
%H A092246 T. D. Noe, <a href="/A092246/b092246.txt">Table of n, a(n) for n = 1..1000</a>
%H A092246 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A092246 <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F A092246 a(n) = 4*n + 2*A010060(n-1) - 3;
%F A092246 a(n) = 2*A001969(n-1) + 1.
%t A092246 Table[If[n < 1, 0, 2 n - 1 - Mod[First@ DigitCount[n - 1, 2], 2]], {n, 120}] /. n_ /; EvenQ@ n -> Nothing (* _Michael De Vlieger_, Feb 29 2016 *)
%t A092246 Select[Range[1, 1001, 2], OddQ[Total[IntegerDigits[#, 2]]]&] (* _Jean-François Alcover_, Mar 15 2016 *)
%o A092246 (PARI) is(n)=n%2&&hammingweight(n)%2 \\ _Charles R Greathouse IV_, Mar 21 2013
%o A092246 (PARI) a(n)=4*n-if(hammingweight(n-1)%2,1,3) \\ _Charles R Greathouse IV_, Mar 22 2013
%o A092246 (Haskell)
%o A092246 a092246 n = a092246_list !! (n - 1)
%o A092246 a092246_list = filter odd a000069_list
%o A092246 -- _Reinhard Zumkeller_, Oct 28 2013
%o A092246 (Python)
%o A092246 def A092246(n): return (n<<2)-(1 if (n-1).bit_count()&1 else 3) # _Chai Wah Wu_, Mar 03 2023
%Y A092246 Cf. A129771, A026147.
%Y A092246 Cf. A230709 (complement).
%Y A092246 Cf. A268717, A268718, A268673.
%K A092246 nonn,easy
%O A092246 1,2
%A A092246 _Benoit Cloitre_, Feb 23 2004