Search: id:A087003
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%I A087003
%S A087003 1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,
               0,1,
%T A087003 0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,
               1,0,0,
%U A087003 0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,
               1,0,1
%V A087003 1,0,-1,0,-1,0,-1,0,0,0,-1,0,-1,0,1,0,-1,0,-1,0,1,0,-1,0,0,0,0,0,-1,0,
               -1,0,1,0,1,0,-1,
%W A087003 0,1,0,-1,0,-1,0,0,0,-1,0,0,0,1,0,-1,0,1,0,1,0,-1,0,-1,0,0,0,1,0,-1,0,
               1,0,-1,0,-1,0,0,
%X A087003 0,1,0,-1,0,0,0,-1,0,1,0,1,0,-1,0,1,0,1,0,1,0,-1,0,0,0,-1,0,-1,0,-1,0,
               -1,0,-1,0,1,0,-1
%N A087003 Sum of Moebius-function values computed for terms of 3x+1 trajectory 
               started at n.
%C A087003 Observe that (these summatory) terms are from {-1,0,1}, so behave like 
               Moebius-function values not like Mertens- function values. Moreover,
               empirically: a(n) deviates from mu[initial-value]=mu[n] only if iv=n 
               is an even square-free number(i,e, it is from A039956).
%C A087003 Comments from Marc LeBrun, Feb 19, 2004: "Absolute values are the same 
               as those of A091069. First consider the descending parts of Collatz 
               (or 3x+1) trajectories, those that begin with even numbers 2^p k, 
               with k odd. These go 2^p k, 2^p-1 k, ... 2k, k. All but 2k and k 
               are divisible by 4, a (rational) square, hence their mu values are 
               all 0 and so they contribute nothing to the sum.
%C A087003 "Then at the end, since mu(2k) = -mu(k), the last two steps cancel each 
               other out. So every descending chain in a trajectory contributes 
               0. Of course the full trajectory of every even number consists entirely 
               of descending chains, so A087003 is 0 for all even n.
%C A087003 "On the other hand, the trajectory of every odd number consists of just 
               that number followed by the trajectory of an even number (which contributes 
               nothing) so A087003 is indeed equal to mu(n) for odd n."
%C A087003 The sequence is multiplicative; it may be defined as the Dirichlet inverse 
               of the integers modulo 2 (A000035). - Gerard P. Michon (g.michon(AT)att.net), 
               Apr 29 2007
%C A087003 a(n) appears in the second column of A156241 at every second row. [From 
               Mats Granvik (mats.granvik(AT)abo.fi), Feb 07 2009]
%H A087003 G. P. Michon, <a href="http://www.numericana.com/answer/open.htm#collatz">
               The Collatz problem</a>.
%H A087003 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">
               Multiplicative functions</a>.
%t A087003 c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Delete[FixedPointList[c, 
               x], -1] lf[x_] := Length[fpl[x]] Table[Apply[Plus, Table[MoebiusMu[Part[fpl[w], 
               j]], {j, 1, lf[w]}]], {w, 1, 256}]
%Y A087003 Cf. A039956.
%Y A087003 Cf. A000035 (the Dirichlet inverse).
%Y A087003 Sequence in context: A147850 A099991 A091069 this_sequence A104606 A014389 
               A014349
%Y A087003 Adjacent sequences: A087000 A087001 A087002 this_sequence A087004 A087005 
               A087006
%K A087003 sign,mult
%O A087003 1,1
%A A087003 Labos E. (labos(AT)ana.sote.hu), Oct 02 2003

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