A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 7, 1, 3, 3, 35, 1, 7, 1, 7, 3, 3, 1, 11, 3, 3, 5, 7, 1, 3, 1, 63, 3, 3, 3, 9, 1, 3, 3, 11, 1, 3, 1, 7, 7, 3, 1, 75, 3, 7, 3, 7, 1, 11, 3, 11, 3, 3, 1, 1, 1, 3, 7, 231, 3, 3, 1, 7, 3, 3, 1, 19, 1, 3, 7, 7, 3, 3, 1, 75, 35, 3, 1, 1, 3, 3, 3, 11, 1, 1, 3, 7, 3, 3, 3, 133, 1, 7, 7, 9, 1, 3, 1, 11, 3

Offset: 1

Author

Antti Karttunen, Aug 12 2018

Keywords

• sign
• frac

Comments

The first negative term is a(210) = -7.

Links

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

Crossrefs

Cf. A001221, A063524, A046644 (denominators).

Cf. also A317831, A317925, A317933, A317845, A317846, A317936, A317938, A317939.

Programs

PARI

A317937aux(n) = if(1==n,n,(omega(n)-sumdiv(n,d,if((d>1)&&(dA317937aux(d)*A317937aux(n/d),0)))/2);
A317937(n) = numerator(A317937aux(n));

PARI

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
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&&dAndrew Howroyd, Aug 13 2018

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001221(n) - Sum_{d|n, d>1, d 1.