

A123087


Sequence of numbers such that a(2*n) + a(n) = n and a(n) is the smallest number such that a(n) >= a(n1).


3



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
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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^k1)/3) = (4^k1)/9  k/3, i.e., limsupa(n)  n/3 = infinity.  Vladimir Shevelev, Aug 25 2010
a(n) = (n  A065359(n))/3.  Velin Yanev, Jul 13 2021


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

Cf. A096268.
Partial sums of A328979.
Sequence in context: A102515 A276571 A066063 * A071868 A179390 A237819
Adjacent sequences: A123084 A123085 A123086 * A123088 A123089 A123090


KEYWORD

nonn


AUTHOR

Philippe Deléham, Sep 27 2006


STATUS

approved



