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A349913 Sum of A001227 (the number of odd divisors function) and its Dirichlet inverse.

%I A349913 #12 Dec 10 2021 22:55:48
%S A349913 2,0,0,1,0,4,0,1,4,4,0,2,0,4,8,1,0,2,0,2,8,4,0,2,4,4,4,2,0,0,0,1,8,4,
%T A349913 8,3,0,4,8,2,0,0,0,2,4,4,0,2,4,2,8,2,0,4,8,2,8,4,0,4,0,4,4,1,8,0,0,2,
%U A349913 8,0,0,3,0,4,4,2,8,0,0,2,5,4,0,4,8,4,8,2,0,8,8,2,8,4,8,2,0,2,4,3,0,0,0,2,0
%N A349913 Sum of A001227 (the number of odd divisors function) and its Dirichlet inverse.
%H A349913 Antti Karttunen, <a href="/A349913/b349913.txt">Table of n, a(n) for n = 1..16384</a>
%F A349913 a(n) = A001227(n) + A327276(n).
%F A349913 a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A001227(d) * A327276(n/d).
%F A349913 a(4*n) = A001227(n).
%t A349913 f1[p_,e_] := If[p==2, 1, e+1]; f2[p_, e_] := Which[e == 1, -1 - Boole[p > 2], e == 2, Boole[p > 2], e > 2, 0]; a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 100] (* _Amiram Eldar_, Dec 08 2021 *)
%o A349913 (PARI)
%o A349913 A001227(n) = numdiv(n>>valuation(n, 2));
%o A349913 A327276(n) = sumdiv(n, d, if(d%2, moebius(d)*moebius(n/d))); \\ From A327276
%o A349913 A349913(n) = (A001227(n)+A327276(n));
%Y A349913 Cf. A001227 (also a quadrisection of this sequence), A327276.
%Y A349913 Cf. also A349914, A349916.
%K A349913 nonn
%O A349913 1,1
%A A349913 _Antti Karttunen_, Dec 08 2021