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

Author

Antti Karttunen, Aug 31 2018

Keywords

• sign
• frac
• mult

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

Crossrefs

Cf. A046644 or A318658 (denominators).

Cf. also A087003, A257098, A318659.

Programs

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&&dA087003(n)));
A318657(n) = numerator(v318657_18[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A087003(n) - Sum_{d|n, d>1, d 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