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A357698 a(n) is the sum of the aliquot divisors of n that are cubefree.

%I A357698 #10 Oct 12 2022 04:36:13
%S A357698 0,1,1,3,1,6,1,7,4,8,1,16,1,10,9,7,1,21,1,22,11,14,1,28,6,16,13,28,1,
%T A357698 42,1,7,15,20,13,55,1,22,17,42,1,54,1,40,33,26,1,28,8,43,21,46,1,39,
%U A357698 17,56,23,32,1,108,1,34,41,7,19,78,1,58,27,74,1,91,1,40
%N A357698 a(n) is the sum of the aliquot divisors of n that are cubefree.
%H A357698 Amiram Eldar, <a href="/A357698/b357698.txt">Table of n, a(n) for n = 1..10000</a>
%F A357698 a(n) = Sum_{d|n, d<n} A212793(d)*d.
%F A357698 a(n) = A073185(n) - (A212793(n)*n).
%F A357698 a(n) = 1 iff n is a prime.
%F A357698 Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2) - 1)/(2*zeta(3)) = 0.268262... .
%e A357698 The divisors of 16 that are cubefree are {1, 2, 4}, and their sum is a(16) = 1 + 2 + 4 = 7.
%t A357698 f[p_, e_] := 1 + p + If[e == 1, 0, p^2]; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;;, 2]], # < 3 &], n, 0]; Array[a, 100]
%o A357698 (PARI) a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, 1 + f[i,1] + if(f[i,2] == 1, 0, f[i,1]^2)); if(n==1 || vecmax(f[,2]) < 3, s -= n); s};
%Y A357698 Cf. A004709, A073185, A212793, A293228.
%Y A357698 Cf. A013661, A002117.
%K A357698 nonn
%O A357698 1,4
%A A357698 _Amiram Eldar_, Oct 10 2022