A318667 Numerators of the sequence whose Dirichlet convolution with itself yields A318307, which is multiplicative with A318307(p^e) = 2^A002487(e).

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, -43, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, -5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Author

Antti Karttunen, Sep 03 2018

Keywords

• sign
• frac
• mult

Comments

Multiplicative because A318307 and A317934 are.

Links

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

Crossrefs

Cf. A318307, A317934 (denominators).

Programs

PARI

up_to = 1+(2^16);
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&&dA002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318307(n) = factorback(apply(e -> 2^A002487(e),factor(n)[,2]));
v318667_aux = DirSqrt(vector(up_to, n, A318307(n)));
A318667(n) = numerator(v318667_aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A318307(n) - Sum_{d|n, d>1, d 1.