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%I #34 Apr 28 2022 13:10:35
%S 0,1,-1,0,1,0,-1,0,0,1,1,0,-1,0,0,1,1,0,-1,0,-1,1,1,-1,0,0,0,0,-1,1,1,
%T 0,0,1,0,0,-1,0,-1,1,1,-1,-1,0,0,1,1,0,0,0,0,0,-1,1,1,-1,-1,0,1,0,-1,
%U 1,0,0,0,1,1,0,0,1,-1,0,1,0,0,0,0,-1,-1,1,-1,1,1,0,1,0,-1,1,-1,0,-1,0,0,1,0,0,1,0,0,0,-1,1,1,-1,-1
%N Inverse Möbius transform of A332823.
%C a(n) is determined by the cubefree part of n, and has the range {-1, 0, 1}.
%C Proof: A332823 is the scaled imaginary part of a completely multiplicative function, f, from the positive integers to the Eisenstein integers (the range of f being the cube roots of unity). Let g be the inverse Moebius transform of f, which is therefore multiplicative. As a function, "scaling the imaginary part" is a homomorphism with respect to addition, so (a(n)) -- being the inverse Moebius transform of A332823 -- is a scaled imaginary part of g. We can show the range of g is the 7 Eisenstein integers closest to 0, namely the 6 sixth roots of unity and 0 itself. We deduce (a(n)) has the range {-1, 0, 1} (in contrast to say, A353364).
%C See A353446, which is twice the real part of g, for further details.
%H Antti Karttunen, <a href="/A353354/b353354.txt">Table of n, a(n) for n = 1..65537</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinInteger.html">Eisenstein Integer</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(n) = Sum_{d|n} A332823(d).
%F a(n) = A353328(n) - A353329(n) = A353328(n) - A353328(A003961(n)).
%F a(n) = A008966(m) * A128834(A090882(m)) = A008966(m) * A128834(A195017(m) mod 6), where m = A050985(n), the cubefree part of n, and A008966(.) is the characteristic function of squarefree numbers.
%F For all n >= 1, a(A003961(n)) = -a(n); and for all m >= 1, a(n*m^3) = a(n).
%o (PARI)
%o 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 A353354(n) = sumdiv(n,d,A332823(d));
%Y Sequences used in a formula defining this sequence: A008966, A050985, A090882, A128834, A195017, A332823, A353328, A353329.
%Y Cf. A003961, A048675, A353352, A353446.
%Y Positions of particular values: A353355 (0), A353356 (1), A353357 (-1).
%Y Somewhat analogous sequence: A353364.
%K sign
%O 1
%A _Antti Karttunen_ and _Peter Munn_, Apr 15 2022
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