A349916 - OEIS

%I #16 Dec 10 2021 22:55:17

%S 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 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 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 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 Antti Karttunen, <a href="/A349916/b349916.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = A113415(n) + A349915(n).

%F a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A113415(d) * A349915(n/d).

%F For all n >= 1, a(4*n) = A113415(n).

%t 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 (PARI)

%o A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));

%o memoA349915 = Map();

%o 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(n) = (A113415(n)+A349915(n));

%Y Cf. A113415 (also a quadrisection of this sequence), A349915.

%Y Cf. also A349913, A349914.

%K nonn

%O 1,1

%A _Antti Karttunen_, Dec 07 2021