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%I #13 Nov 02 2022 07:48:01
%S 1,3,7,14,22,33,46,60,76,95,116,138,163,189,217,248,280,315,352,390,
%T 431,473,517,564,613,663,715,770,826,885,946,1008,1072,1139,1208,1278,
%U 1351,1425,1501,1580,1661,1743,1827,1914,2002,2093,2186,2280,2377,2475,2575
%N Partial sums of A000069.
%C Partial sums of odious numbers. Second differences give A007413. The subsequence of prime partial sums of odious numbers begins: 3, 7, 163, 431, 613, 2377, 3691, which is a subsequence of A027697. The subsequence of odious partial sums of odious numbers begins: 1, 7, 14, 22, 76, 138, 217, 280, 352, 431, 517, 613, 770, 885.
%H Michael De Vlieger, <a href="/A173209/b173209.txt">Table of n, a(n) for n = 1..16384</a>
%H Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, <a href="http://arxiv.org/abs/1405.6214">Beyond odious and evil</a>, arXiv preprint arXiv:1405.6214 [math.NT], 2014.
%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 52.
%F a(n) = SUM[i=1..n] A000069(i) = SUM[i=1..n] {i such that A010060(i)=1}.
%F a(n) = n^2 - n/2 + O(1). - _Charles R Greathouse IV_, Mar 22 2013
%e a(65) = 1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 + 16 + 19 + 21 + 22 + 25 + 26 + 28 + 31 + 32 + 35 + 37 + 38 + 41 + 42 + 44 + 47 + 49 + 50 + 52 + 55 + 56 + 59 + 61 + 62 + 64 + 67 + 69 + 70 + 73 + 74 + 76 + 79 + 81 + 82 + 84 + 87 + 88 + 91 + 93 + 94 + 97 + 98 + 100 + 103 + 104 + 107 + 109 + 110 + 112 + 115 + 117 + 118 + 121 + 122 + 124 + 127 + 128.
%t Accumulate@ Select[Range[100], OddQ@ DigitCount[#, 2, 1] &] (* _Michael De Vlieger_, Nov 01 2022 *)
%o (PARI) a(n)=n^2-(n+1)\2+(n%2&&hammingweight(n)%2) \\ _Charles R Greathouse IV_, Mar 22 2013
%Y Cf. A000069, A000079, A000120, A000773, A000788, A001969, A002042, A002089, A007413, A019568, A023416, A027697, A059015, A083420, A010060, A132679, A133009.
%K base,easy,nonn
%O 1,2
%A _Jonathan Vos Post_, Feb 12 2010
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