A349914 - OEIS

%I #11 Dec 10 2021 22:55:39

%S 2,0,0,1,0,8,0,1,16,12,0,4,0,16,48,1,0,10,0,6,64,24,0,4,36,28,40,8,0,

%T 0,0,1,96,36,96,13,0,40,112,6,0,0,0,12,60,48,0,4,64,26,144,14,0,40,

%U 144,8,160,60,0,24,0,64,80,1,168,0,0,18,192,0,0,13,0,76,104,20,192,0,0,6,121,84,0,32,216,88,240

%N Sum of A000593 (the sum of odd divisors function) and its Dirichlet inverse.

%H Antti Karttunen, <a href="/A349914/b349914.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = A000593(n) + A327278(n).

%F a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A000593(d) * A327278(n/d).

%F a(4*n) = A000593(n).

%t f1[p_,e_] := If[p==2, 1, (p^(e+1)-1)/(p-1)]; f2[p_, e_] := If[p == 2, -Boole[e == 1], Which[e == 1, -p - 1, e == 2, p, 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 (PARI)

%o A000593(n) = sigma(n>>valuation(n, 2));

%o A327278(n) = sumdiv(n,d,if(d%2,d*moebius(d)*moebius(n/d),0));

%o A349914(n) = (A000593(n)+A327278(n));

%Y Cf. A000593 (also a quadrisection of this sequence), A327278.

%Y Cf. also A349913, A349916.

%K nonn

%O 1,1

%A _Antti Karttunen_, Dec 08 2021