The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

Antti Karttunen, <a href="/A318658/b318658.txt">Table of n, a(n) for n = 1..16384</a>

Denominators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

a(n) = 2^A318659(n).

allocated for Antti KarttunenDenominators of the sequence whose Dirichlet convolution with itself yields A087003.

1, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 8, 1, 16, 1, 2, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 16, 1, 2, 1, 8, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 16, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 16, 1, 4, 1, 2, 1, 128, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 4, 1, 2, 1, 16, 1, 2, 1, 2, 1, 8

1,3

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A087003(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

(PARI)

up_to = 65537;

A087003(n) = ((n%2)*moebius(n)); \\ I.e. a(n) = A000035(n)*A008683(n).

DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u};

v318657_18 = DirSqrt(vector(up_to, n, A087003(n)));

A318657(n) = numerator(v318657_18[n]);

A318658(n) = denominator(v318657_18[n]);

Cf. A087003, A318657, A318659.

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Antti Karttunen, Aug 31 2018

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