A091069 -id:A091069 - cp's OEIS frontend

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

Labos Elemer, Oct 02 2003

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

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

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

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

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 *)

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

a(n)={sumdiv(n, d,  my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 04 2018

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

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

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

A318608 Moebius function mu(n) defined for the Gaussian integers.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Aug 30 2018

Just like the original Moebius function over the integers, a(n) = 0 if n has a squared Gaussian prime factor, otherwise (-1)^t if n is a product of a Gaussian unit and t distinct Gaussian prime factors.
a(n) = 0 for even n since 2 = -i*(1 + i)^2 contains a squared factor. For rational primes p == 1 (mod 4), p is always factored as (x + y*i)(x - y*i), x + y*i and x - y*i are not associated so a(p) = (-1)*(-1) = 1.
Interestingly, a(n) and A091069(n) have the same absolute value (= |A087003(n)|), since the discriminants of the quadratic fields Q[i] and Q[sqrt(2)] are -4 and 8 respectively, resulting in Q[i] and Q[sqrt(2)] being two of the three quadratic fields with discriminant a power of 2 or negated (the other one being Q[sqrt(-2)] with discriminant -8).

			a(15) = -1 because 15 is factored as 3*(2 + i)*(2 - i) with three distinct Gaussian prime factors.
a(21) = (-1)*(-1) = 1 because 21 = 3*7 where 3 and 7 are congruent to 3 mod 4 (thus being Gaussian primes).

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Gaussian integer

Absolute values are the same as those of A087003.
First row and column of A103226.
Cf. A101455.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), this sequence ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319448.
Cf. A091069 (Moebius function over Z[sqrt(2)]).

f[p_, e_] := If[p == 2 || e > 1, 0, Switch[Mod[p, 4], 1, 1, 3, -1]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2020 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2||e>=2, r=0);
        if(Mod(p,4)==3&e==1, r*=-1);
    );
    return(r);
}

a(n) = 0 if n even or has a square prime factor, otherwise Product_{p divides n} (2 - (p mod 4)) where the product is taken over the primes.
Multiplicative with a(p^e) = 0 if p = 2 or e > 1, a(p) = 1 if p == 1 (mod 4) and -1 if p == 3 (mod 4).
a(n) = 0 if A078458(n) != A086275(n), otherwise (-1)^A086275(n).
a(n) = A103226(n,0) = A103226(0,n).
For squarefree n, a(n) = Kronecker symbol (-4, n) = A101455(n). Also for these n, a(n) = A091069(n) if n even or n == 1 (mod 8), otherwise -A091069(n).

cp's OEIS Frontend

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.

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A318608 Moebius function mu(n) defined for the Gaussian integers.

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