

A087003


a(2n) = 0 and a(2n+1) = mu(2n+1); also the sum of Mobius function values computed for terms of 3x+1 trajectory started at n, provided that Collatzconjecture is true.


5



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, 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, 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
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OFFSET

1,1


COMMENTS

Observe that (these summatory) terms are from {1,0,1}, so behave like Mobius function values, not like Mertens function values. Moreover, empirically: a(n) deviates from mu(initialvalue) = mu(n) only if iv = n is an even squarefree number (i.e., it is from A039956).  This comment, like also the next one, concerns the original Collatzrelated definition of this sequence.  Antti Karttunen, Sep 18 2017
From Marc LeBrun, Feb 19 2004: (Start)
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^p1 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.
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.
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.
(End)
The sequence is multiplicative; it may be defined as the Dirichlet inverse of the integers modulo 2 (A000035).  Gerard P. Michon, Apr 29 2007
a(n) appears in the second column of A156241 at every second row.  Mats Granvik, Feb 07 2009


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000
G. P. Michon, The Collatz problem.
G. P. Michon, Multiplicative functions.
Index entries for sequences related to 3x+1 (or Collatz) problem


FORMULA

a(n) = A008683(n) + A292273(n).  Antti Karttunen, Sep 14 2017


MATHEMATICA

c[x_] := (1Mod[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}]


PROG

(PARI)
A006370(n) = if(n%2, 3*n+1, n/2); \\ This function from Michael B. Porter, May 29 2010
A087003(n) = { my(s=1); while(n>1, s += moebius(n); n = A006370(n)); (s); }; \\ Antti Karttunen, Sep 14 2017


CROSSREFS

Cf. A006370, A008683, A039956, A292273, A099990.
Cf. A000035 (the Dirichlet inverse).
Sequence in context: A080545 A099991 A091069 * A104606 A014389 A014349
Adjacent sequences: A087000 A087001 A087002 * A087004 A087005 A087006


KEYWORD

sign,mult


AUTHOR

Labos Elemer, Oct 02 2003


EXTENSIONS

a(2n) = 0, a(2n+1) = mu(2n+1) added to the name as the new primary definition by Antti Karttunen, Sep 18 2017


STATUS

approved



