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a(n) = A349444(n) + A349445(n).
4

%I #10 Nov 21 2021 01:19:53

%S 2,0,0,1,0,-2,0,1,1,-4,0,-1,0,-6,4,1,0,-5,0,-2,6,-10,0,-1,4,-12,5,-3,

%T 0,-4,0,1,10,-16,12,-2,0,-18,12,-2,0,-6,0,-5,14,-22,0,-1,9,-16,16,-6,

%U 0,-13,20,-3,18,-28,0,0,0,-30,21,1,24,-10,0,-8,22,-12,0,-2,0,-36,24,-9,30,-12,0,-2,19,-40,0,0,32

%N a(n) = A349444(n) + A349445(n).

%H Antti Karttunen, <a href="/A349446/b349446.txt">Table of n, a(n) for n = 1..20000</a>

%F a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A349444(d) * A349445(n/d). [As the sequences are Dirichlet inverses of each other]

%t s[n_] := MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#]*k[n/#] &, # < n &]; a[n_] := DivisorSum[n, s[#]*k[n/#] + IntegerExponent[2*#, 2]*kinv[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2021 *)

%o (PARI) A349446(n) = (A349444(n)+A349445(n)); \\ Needs also code from A349444 and A349445.

%Y Cf. A349444, A349445.

%Y Cf. also A349433.

%K sign

%O 1,1

%A _Antti Karttunen_, Nov 18 2021