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.

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

Offset: 1

Author

Labos Elemer, Oct 02 2003

Keywords

• sign
• mult

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). - This comment, like also the next one, concerns the original Collatz-related 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^(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

• 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

Crossrefs

Cf. A006370, A008683, A039956, A292273, A099990, A209229.

Cf. A000035 (the Dirichlet inverse), A318657/A318658 (the "Dirichlet Square Root").

Programs

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}]
Riffle[MoebiusMu[Range[1,121,2]],0] (* Harvey P. Dale, Jan 24 2025 *)

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<Andrew Howroyd, Aug 04 2018

PARI

A087003(n) = ((n%2)*moebius(n)); \\ Antti Karttunen, Sep 01 2018

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)

Sum_{n>=1} a(n)/n^2 = A217739 . Sum_{n>=1} a(n)/n^3 = A233091. Sum_{n>=1} a(n)/n^4 = A300707. - R. J. Mathar, Dec 17 2024

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