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A349916 Sum of A113415 and its Dirichlet inverse, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

%I A349916 #16 Dec 10 2021 22:55:17
%S A349916 2,0,0,1,0,6,0,1,9,8,0,3,0,10,24,1,0,7,0,4,30,14,0,3,16,16,21,5,0,4,0,
%T A349916 1,42,20,40,8,0,22,48,4,0,6,0,7,40,26,0,3,25,18,60,8,0,23,56,5,66,32,
%U A349916 0,14,0,34,53,1,64,10,0,10,78,12,0,8,0,40,70,11,70,12,0,4,61,44,0,18,80,46,96,7,0,44,80
%N A349916 Sum of A113415 and its Dirichlet inverse, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.
%H A349916 Antti Karttunen, <a href="/A349916/b349916.txt">Table of n, a(n) for n = 1..16384</a>
%F A349916 a(n) = A113415(n) + A349915(n).
%F A349916 a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A113415(d) * A349915(n/d).
%F A349916 For all n >= 1, a(4*n) = A113415(n).
%t A349916 s[n_] := DivisorSum[n, (# + 1) * Mod[#, 2] &] / 2; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := s[n] + sinv[n]; Array[a, 100] (* _Amiram Eldar_, Dec 08 2021 *)
%o A349916 (PARI)
%o A349916 A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
%o A349916 memoA349915 = Map();
%o A349916 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)));
%o A349916 A349916(n) = (A113415(n)+A349915(n));
%Y A349916 Cf. A113415 (also a quadrisection of this sequence), A349915.
%Y A349916 Cf. also A349913, A349914.
%K A349916 nonn
%O A349916 1,1
%A A349916 _Antti Karttunen_, Dec 07 2021