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%I #33 Mar 05 2023 11:30:39
%S 0,0,1,1,1,1,2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,9,9,9,10,10,11,
%T 11,12,12,12,12,13,13,14,14,15,15,15,15,16,16,16,16,17,17,17,17,18,18,
%U 19,19,20,20,20,20,21,21,21,21,22,22,22,22,23,23,24,24,25,25,25,25,26
%N Sequence of numbers such that a(2*n) + a(n) = n and a(n) is the smallest number such that a(n) >= a(n-1).
%C If the value a(n) = m >= 1 is appearing for the first time, then n is of the form n = 2^k*s, where k,s are odd numbers. Therefore every m occurs 2 or 4 times consecutively. More exactly, if n+2 has the same form as n (i.e., 2^k*s with odd k,s), then a(n) = m occurs 2 times, otherwise, m occurs 4 times. - _Vladimir Shevelev_, Aug 25 2010
%C a(n) is the number of those numbers not exceeding n for which 2 is an infinitary divisor (for definition see comment at A037445). - _Vladimir Shevelev_, Feb 21 2011
%H Reinhard Zumkeller, <a href="/A123087/b123087.txt">Table of n, a(n) for n = 0..10000</a>
%F a(0)=0, a(n) = floor(n/2) - a(floor(n/2)); partial sums of A096268; a(2n) = A050292(n); a(n) is asymptotic to n/3. - _Benoit Cloitre_, Sep 30 2006
%F a(2*n+1) = a(2*n); a(n) = n/3 + O(log(n)), moreover, the equation a(3m) = m has infinitely many solutions, e.g., a(3*2^k) = 2^k; on the other hand, a((4^k-1)/3) = (4^k-1)/9 - k/3, i.e., limsup|a(n) - n/3| = infinity. - _Vladimir Shevelev_, Aug 25 2010
%F a(n) = (n - A065359(n))/3. - _Velin Yanev_, Jul 13 2021
%F a(n) = n - A050292(n). - _Max Alekseyev_, Mar 05 2023
%e a(2*0) + a(0) = 0 -----> a(0) = 0
%e a(1) >= a(0) ---------> a(1) = 0
%e a(2*1) + a(1) = 1 -----> a(2) = 1
%e a(3) >= a(2) ---------> a(3) = 1
%e a(2*2) + a(2) = 2 -----> a(4) = 1
%e a(5) >= a(4) ---------> a(5) = 1
%e a(2*3) + a(3) = 3 -----> a(6) = 2
%e a(7) >= a(6) ---------> a(7) = 2
%e a(2*4) + a(4) = 4 -----> a(8) = 3
%e a(9) >= a(8) ---------> a(9) = 3
%e a(2*5) + a(5) = 5 -----> a(10) = 4
%e a(11) >= a(10) --------> a(11) = 4
%e a(2*6) + a(6) = 6 -----> a(12) = 4
%e a(13) >= a(12) --------> a(13) = 4
%e a(2*7) + a(7) = 7 -----> a(14) = 5
%o (PARI) a(n)=if(n<1,0,floor(n/2)-a(floor(n/2))) \\ _Benoit Cloitre_, Sep 30 2006
%o (Haskell)
%o a123087 n = a123087_list !! n
%o a123087_list = scanl (+) 0 a096268_list
%o -- _Reinhard Zumkeller_, Jul 29 2014
%Y Partial sums of A096268 and of A328979.
%Y Cf. A050292, A065359.
%K nonn
%O 0,7
%A _Philippe Deléham_, Sep 27 2006
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