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A087003 Sum of Mobius function values computed for terms of 3x+1 trajectory started at n. 4
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 (list; graph; refs; listen; history; text; internal format)
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(initial-value) = mu(n) only if iv = n is an even squarefree number (i.e., it is from A039956).

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^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.

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

Table of n, a(n) for n=1..113.

G. P. Michon, The Collatz problem.

G. P. Michon, Multiplicative functions.

MATHEMATICA

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}]

CROSSREFS

Cf. A039956.

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

STATUS

approved

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Last modified August 17 06:03 EDT 2017. Contains 290635 sequences.