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These are also denominators (in their lowest terms) for the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.

From Antti Karttunen, Sep 02 2018: (Start)

Proof for the above claim:

This sequence is defined as the denominator (given in the lowest terms) of rational valued function r(1) = 1, r(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. Define sequence Ay(n) as the denominator of function s(n), with otherwise similar definition, but with A064549 in place of A000290. Let Ay(n) be the denominator of s(n), reduced also into the lowest terms. (Corresponding numerators are A318649 and A318511 respectively. Note that the denominators in both cases must always be of the form 2^k, with k >= 0).

By applying the distributive property of Dirichlet Convolution [which says that for any completely multiplicative function f, it doesn't matter whether one multiplies the result of convolution afterwards, or whether one multiplies the operands separately before convolution: f(g * g) = (fg) * (fg)], with A000027 in the role of f in both cases, one obtains a pair of equations:

A318649(n) A318681(n) n*A299149(n)

---------- = ---------- = ------------

A318512(n) A299150(n) A299150(n)

and

A318511(n) A318680(n) n*A318653(n)

---------- = ---------- = ------------

Ay(n) A299150(n) A299150(n)

where the leftmost ratios are reduced into their lowest terms.

Sequence A318656 gives the 2-adic valuation of ratio A318649(n)/A318512(n), and because there are no even terms neither in A299149 nor in A318653, it also gives the 2-adic valuation of the latter ratio. As A318511/Ay is given in the lowest terms (not both of A318511(n) and Ay(n) can be even at same n), this implies that Ay must indeed be identical to A318512, and furthermore that A318655(n) = A007814(A318649(n)) = A007814(A318511(n)).

(End)

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

Because the corresponding denominator sequence A318658 is equal to A046644 on all odd n, and this sequence as well as A087003 is zero on all even n, it means that also the Dirichlet convolution of a(n)/A046644(n) with itself will yield A087003. Because both A046644 and A087003 are multiplicative, this sequence is also. - Antti Karttunen, Sep 01 2018

Denominators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-4)^(1/2).

First differs from A318658 at n = 54.