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%I #18 May 03 2022 13:07:14
%S 2,1,1,0,1,2,1,2,0,-1,1,0,1,2,2,1,1,0,1,0,-1,-1,1,1,0,2,2,0,1,1,1,0,2,
%T -1,2,0,1,2,-1,1,1,1,1,0,0,-1,1,2,0,0,2,0,1,1,-1,1,-1,2,1,0,1,-1,0,2,
%U 2,1,1,0,2,1,1,0,1,2,0,0,2,1,1,-1,1,-1,1,0,-1,2,-1,1,1,0,-1,0,2,-1,2,0,1,0,0,0,1
%N Let g be the inverse Möbius transform of the Eisenstein integer-valued function f defined in A353445. a(n) is twice the real part of g(n).
%C The imaginary part of g(n) is A353354(n)*(sqrt(3)/2)*i.
%C f(n), g(n), and so also a(n), are determined by the cubefree part of n, A050985(n). If the cubefree part is not squarefree, g(n) is 0; otherwise g(n) = x^(A195017(A050985(n)), where x = (1 + sqrt(3)*i)/2, the primitive 6th root of unity with positive imaginary part.
%C The above formula arises from g being multiplicative (because f is multiplicative). g(prime(m)^k) is 1 for k == 0 (mod 3), 0 for k == 2 (mod 3), and for k == 1 (mod 3) the result depends on the parity of m. g(prime(m)^(3k+1)) is 1+w for odd m, -w for even m, where w is the cube root of unity with positive imaginary part. 1+w and -w are the primitive 6th roots of unity.
%C So the range of g is the 6 sixth roots of unity and 0 itself: these are the 7 Eisenstein integers closest to 0, and they are clearly closed under multiplication. The range of (a(n)) is [-2..2]. g(n) and a(n) are 0 if and only if the cubefree part of n is not squarefree. (Compare with the Moebius function being 0 when its argument is not squarefree.) Otherwise a(n) is even if and only if n is in A332820.
%H Peter Munn, <a href="/A353446/b353446.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Munn, <a href="/A353446/a353446.png">Figure showing relationship of the Eisenstein integer g(n) to the presence of n in other sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinInteger.html">Eisenstein Integer</a>
%F a(n) = A008966(m) * A087204(A090882(m)) = A008966(m) * A087204(|A195017(m)|), where m = A050985(n), the cubefree part of n, and A008966(.) is the characteristic function of squarefree numbers.
%F For n >= 1, -2 <= a(n) <= 2.
%F {n : a(n) = -2} = {A211338} INTERSECT {A332820}.
%F {n : a(n) = -1} = {A211337} \ {A332820}.
%F {n : a(n) = 0} = {A059269}.
%F {n : a(n) = 1} = {A211338} \ {A332820}.
%F {n : a(n) = 2} = {A211337} INTERSECT {A332820}.
%o (PARI)
%o A050985(n) = { my(f=factor(n)); f[, 2] = apply(x->(x % 3), f[, 2]); factorback(f); }; \\ From A050985
%o A087204(n) = ([2, 1, -1, -2, -1, 1][1+(n%6)]);
%o A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * (-1)^(1+primepi(f[i,1])))); };
%o A353446(n) = { my(u=A050985(n)); issquarefree(u) * A087204(abs(A195017(u))); };
%Y Sequences used in a definition of this sequence: A008966, A050985, A090882, A087204, A195017, A353445.
%Y See A353354 for the imaginary part.
%Y Positions of particular values depend on A059269, A211337, A211338, A332820 as shown in the formula section.
%Y Positions of even numbers: A353355.
%K sign
%O 1,1
%A _Antti Karttunen_ and _Peter Munn_, Apr 19 2022
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