A318657 Numerators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -5, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1

Offset

1,81

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = numerator 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.

a(2n) = 0, a(2n-1) = A257098(2n-1), thus multiplicative with a(2^e) = 0, a(p^e) = A257098(p^e) for odd primes p. - Antti Karttunen, Sep 01 2018

Prog

(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]);

Crossrefs

Cf. A046644 or A318658 (denominators).

Cf. also A087003, A257098, A318659.

Sequence in context: A113038 A082512 A068385 * A286277 A225749 A227985

Adjacent sequences: A318654 A318655 A318656 * A318658 A318659 A318660

Keyword

sign,frac,mult

Author

Antti Karttunen, Aug 31 2018

Status

approved