

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 Collatz conjecture is true.


12



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
Moebius transform of A209229.  Andrew Howroyd, Aug 04 2018
From Jianing Song, Aug 04 2018: (Start)
Multiplicative with a(2^e) = 0, a(p^e) = (1 + (1)^e)/2 for odd primes p.
Dirichlet g.f.: 1/((1  2^(s))*zeta(s)).
(End)
From Antti Karttunen, Sep 01 2018: (Start)
a(n) = A000035(n)*A008683(n).
Dirichlet convolution of A318657/A046644 with itself.
(End)


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
(PARI) a(n)={sumdiv(n, d, my(e=valuation(d, 2)); if(d==1<<e, moebius(n/d), 0))} \\ Andrew Howroyd, Aug 04 2018
(PARI) A087003(n) = ((n%2)*moebius(n)); \\ Antti Karttunen, Sep 01 2018


CROSSREFS

Cf. A006370, A008683, A039956, A292273, A099990, A209229.
Cf. A000035 (the Dirichlet inverse), A318657/A318658 (the "Dirichlet Square Root").
Sequence in context: A099991 A091069 A318608 * A266840 A104606 A189169
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



