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A359484 a(n) = n * mu(n) if n is odd, otherwise n * mu(n) - (n/2) * mu(n/2).

%I A359484 #32 Jun 09 2023 23:59:03
%S A359484 1,-3,-3,2,-5,9,-7,0,0,15,-11,-6,-13,21,15,0,-17,0,-19,-10,21,33,-23,
%T A359484 0,0,39,0,-14,-29,-45,-31,0,33,51,35,0,-37,57,39,0,-41,-63,-43,-22,0,
%U A359484 69,-47,0,0,0,51,-26,-53,0,55,0,57,87,-59,30,-61,93,0,0,65,-99,-67,-34,69,-105,-71,0
%N A359484 a(n) = n * mu(n) if n is odd, otherwise n * mu(n) - (n/2) * mu(n/2).
%H A359484 Seiichi Manyama, <a href="/A359484/b359484.txt">Table of n, a(n) for n = 1..10000</a>
%F A359484 a(n) = A055615(n) if n is odd, otherwise A055615(n) - A055615(n/2).
%F A359484 a(n) is multiplicative with a(2)= -3, a(4)= 2, a(2^e)= 0 if e>2. a(p)= -p, a(p^e)= 0 if e>1, p>2.
%F A359484 a(1) = 1, a(2) = -3; a(n) = -n * Sum_{d|n, d < n} a(d) / d.
%F A359484 G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k * A(x^k).
%F A359484 a(n) = n*mu(n)-n*mu(n*2^(n mod 2)/2)*((n+1) mod 2)/2. - _Wesley Ivan Hurt_, Jun 09 2023
%t A359484 a[n_] := n * MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]*n/2]; Array[a, 100] (* _Amiram Eldar_, May 09 2023 *)
%o A359484 (PARI) a(n) = n*moebius(n)-if(n%2, 0, n/2*moebius(n/2));
%Y A359484 Partial sums give A359478.
%Y A359484 Cf. A008683, A055615, A358276.
%Y A359484 Cf. A092673, A359485, A359531.
%K A359484 sign,mult
%O A359484 1,2
%A A359484 _Seiichi Manyama_, Mar 31 2023