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A123087
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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).
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1
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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, 11, 12, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 25, 25, 25, 25, 26
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| If value a(n)=m>=1 appears at the first time, then n has the form n=2^k*s, where k,s are odd numbers. Therefore every m repeats 2 or 4 times. More exactly, if n+2 has the same form as n (i.e., 2^k*s with odd k,s), then a(n)=m repeats 2 times, otherwise, m repeats 4 times. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 25 2010]
a(n) is the number of those numbers not exceeding n for which 2 is infinitary divisor (for definition see comment in A037445) - Vladimir Shevelev (shevelev(AT)bgu.ac.il), Feb 21 2011.
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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 (abmt(AT)orange.fr), Sep 30 2006
a(2*n+1)=a(2*n); a(n)=n/3+O(ln(n)), moreover, 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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 25 2010]
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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.
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PROG
| (PARI) a(n)=if(n<1, 0, floor(n/2)-a(floor(n/2))) - Benoit Cloitre (abmt(AT)orange.fr), Sep 30 2006
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CROSSREFS
| Sequence in context: A062571 A102515 A066063 * A071868 A179390 A082447
Adjacent sequences: A123084 A123085 A123086 * A123088 A123089 A123090
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KEYWORD
| nonn
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 27 2006
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