cp's OEIS Frontend

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).

%I A353445 #17 Oct 21 2023 11:34:53
%S A353445 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,
%T A353445 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,
%U A353445 -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
%N 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).
%C A353445 The imaginary part of f(n) is A332823(n)*(sqrt(3)/2)*i.
%C A353445 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).)
%C A353445 The function f is useful for analyzing the inverse Moebius transform of A332823 considered as a stand-alone integer-valued function.
%F A353445 a(n) = 2 - 3 * A332823(n)^2.
%o A353445 (PARI)
%o A353445 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); };
%o A353445 A353445(n) = (2 - (3*(A332823(n)^2)));
%Y A353445 Cf. A048675, A195017, A332823.
%Y A353445 Positions of 2's: A332820.
%Y A353445 For the inverse Moebius transform of f, see A353446.
%K A353445 sign
%O A353445 1,1
%A A353445 _Antti Karttunen_ and _Peter Munn_, Apr 19 2022