A353445 Let f be the completely multiplicative function from the positive integers to the cube roots of unity defined by f(prime(m)) = w^(2^(m-1)), where w is the cube root with positive imaginary part. a(n) is twice the real part of f(n).

2, -1, -1, -1, -1, 2, -1, 2, -1, -1, -1, -1, -1, 2, 2, -1, -1, -1, -1, 2, -1, -1, -1, -1, -1, 2, 2, -1, -1, -1, -1, -1, 2, -1, 2, 2, -1, 2, -1, -1, -1, -1, -1, 2, -1, -1, -1, 2, -1, 2, 2, -1, -1, -1, -1, -1, -1, 2, -1, -1, -1, -1, 2, 2, 2, -1, -1, 2, 2, -1, -1, -1, -1, 2, -1, -1, 2, -1, -1, -1, -1, -1, -1, 2, -1, 2, -1, -1, -1, 2

Offset: 1

Author

Antti Karttunen and Peter Munn, Apr 19 2022

Keywords

• sign

Comments

The imaginary part of f(n) is A332823(n)*(sqrt(3)/2)*i.

f(n) = w^(A048675(n)) = w^(A195017(n)), where w = (-1 + sqrt(3)*i)/2, the primitive cube root of unity with positive imaginary part. (w may also be expressed as e^(i*2*Pi/3).)

The function f is useful for analyzing the inverse Moebius transform of A332823 considered as a stand-alone integer-valued function.

Crossrefs

Cf. A048675, A195017, A332823.

Positions of 2's: A332820.

For the inverse Moebius transform of f, see A353446.

Programs

PARI

A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };
A353445(n) = (2 - (3*(A332823(n)^2)));

Formula

a(n) = 2 - 3 * A332823(n)^2.