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A349915 Dirichlet inverse of A113415, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

%I A349915 #11 Dec 10 2021 22:55:29
%S A349915 1,-1,-3,0,-4,3,-5,0,1,4,-7,0,-8,5,10,0,-10,-1,-11,0,12,7,-13,0,-1,8,
%T A349915 -1,0,-16,-10,-17,0,16,10,14,0,-20,11,18,0,-22,-12,-23,0,-2,13,-25,0,
%U A349915 -5,1,22,0,-28,1,18,0,24,16,-31,0,-32,17,-2,0,20,-16,-35,0,28,-14,-37,0,-38,20,5,0,20,-18,-41,0,-2,22
%N A349915 Dirichlet inverse of A113415, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.
%H A349915 Antti Karttunen, <a href="/A349915/b349915.txt">Table of n, a(n) for n = 1..16384</a>
%F A349915 a(1) = 1; a(n) = -Sum_{d|n, d < n} A113415(n/d) * a(d).
%F A349915 a(n) = A349916(n) - A113415(n).
%t A349915 s[n_] := DivisorSum[n, (# + 1) * Mod[#, 2] &] / 2; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Dec 08 2021 *)
%o A349915 (PARI)
%o A349915 A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
%o A349915 memoA349915 = Map();
%o A349915 A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(d<n,A113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));
%Y A349915 Cf. A000593, A001227, A113415, A349916.
%K A349915 sign
%O A349915 1,3
%A A349915 _Antti Karttunen_, Dec 07 2021