OFFSET
0,7
COMMENTS
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
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
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
FORMULA
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
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
a(n) = (n - A065359(n))/3. - Velin Yanev, Jul 13 2021
a(n) = n - A050292(n). - Max Alekseyev, Mar 05 2023
EXAMPLE
a(2*0) + a(0) = 0 -----> a(0) = 0
a(1) >= a(0) ---------> a(1) = 0
a(2*1) + a(1) = 1 -----> a(2) = 1
a(3) >= a(2) ---------> a(3) = 1
a(2*2) + a(2) = 2 -----> a(4) = 1
a(5) >= a(4) ---------> a(5) = 1
a(2*3) + a(3) = 3 -----> a(6) = 2
a(7) >= a(6) ---------> a(7) = 2
a(2*4) + a(4) = 4 -----> a(8) = 3
a(9) >= a(8) ---------> a(9) = 3
a(2*5) + a(5) = 5 -----> a(10) = 4
a(11) >= a(10) --------> a(11) = 4
a(2*6) + a(6) = 6 -----> a(12) = 4
a(13) >= a(12) --------> a(13) = 4
a(2*7) + a(7) = 7 -----> a(14) = 5
PROG
(PARI) a(n)=if(n<1, 0, floor(n/2)-a(floor(n/2))) \\ Benoit Cloitre, Sep 30 2006
(Haskell)
a123087 n = a123087_list !! n
a123087_list = scanl (+) 0 a096268_list
-- Reinhard Zumkeller, Jul 29 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Deléham, Sep 27 2006
STATUS
approved