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%I #22 Apr 04 2024 07:55:25
%S 0,0,0,2,0,0,0,0,3,0,0,8,0,0,0,4,0,9,0,12,0,0,0,0,5,0,0,16,0,0,0,0,0,
%T 0,0,41,0,0,0,0,0,0,0,24,18,0,0,16,7,15,0,28,0,0,0,0,0,0,0,48,0,0,24,
%U 8,0,0,0,36,0,0,0,45,0,0,20,40,0,0,0,24,9,0,0,64,0,0,0,0,0,54,0,48,0,0,0,0,0,21,36,87
%N a(n) is the sum of the non-bi-unitary divisors of n.
%H Amiram Eldar, <a href="/A319072/b319072.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000203(n) - A188999(n).
%e For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12, and the bi-unitary divisors of 12 are 1, 3, 4, 12, hence the non-bi-unitary divisors of 12 are 2 and 6, and the sum of them is 2 + 6 = 8, so a(12) = 8. Also the sum of the divisors of 12 is 28, and the sum of the bi-unitary divisors of 12 is 20, so a(12) = 28 - 20 = 8.
%t f1[p_, e_] := (p^(e+1) - 1)/(p - 1); f2[p_, e_] := f1[p, e] - If[OddQ[e], 0, p^(e/2)]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100] (* _Amiram Eldar_, Apr 04 2024 *)
%Y Cf. A000203, A027750, A188999, A222266.
%K nonn
%O 1,4
%A _Omar E. Pol_, Sep 22 2018