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A293228 a(n) is the sum of proper divisors of n that are squarefree.

%I A293228 #22 Apr 20 2025 10:17:45
%S A293228 0,1,1,3,1,6,1,3,4,8,1,12,1,10,9,3,1,12,1,18,11,14,1,12,6,16,4,24,1,
%T A293228 42,1,3,15,20,13,12,1,22,17,18,1,54,1,36,24,26,1,12,8,18,21,42,1,12,
%U A293228 17,24,23,32,1,72,1,34,32,3,19,78,1,54,27,74,1,12,1,40,24,60,19,90,1,18,4,44,1,96,23,46,33,36,1,72,21,72,35,50,25,12,1,24
%N A293228 a(n) is the sum of proper divisors of n that are squarefree.
%H A293228 Antti Karttunen, <a href="/A293228/b293228.txt">Table of n, a(n) for n = 1..16384</a>
%F A293228 a(n) = Sum_{d|n, d<n} A008966(d)*d.
%F A293228 a(n) = A048250(n) - (A008966(n)*n).
%F A293228 G.f.: Sum_{k>=1} mu(k)^2*k*x^(2*k)/(1 - x^k). - _Ilya Gutkovskiy_, Oct 28 2018
%F A293228 From _Amiram Eldar_, Oct 09 2022: (Start)
%F A293228 a(n) = 1 iff n is a prime.
%F A293228 a(n) = 3 iff n is a power of 2 greater than 2 (A020707).
%F A293228 Sum_{k=1..n} a(k) ~ (1/2 - 3/Pi^2) * n^2. (End)
%p A293228 with(numtheory): seq(coeff(series(add(mobius(k)^2*k*x^(2*k)/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 120); # _Muniru A Asiru_, Oct 28 2018
%t A293228 a[n_] := Times @@ (1 + (f = FactorInteger[n])[[;; , 1]]) - If[AllTrue[f[[;;, 2]], # == 1 &], n, 0]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Oct 09 2022 *)
%t A293228 Table[Total[Select[Most[Divisors[n]],SquareFreeQ]],{n,100}] (* _Harvey P. Dale_, Apr 20 2025 *)
%o A293228 (PARI) A293228(n) = sumdiv(n, d, (d<n)*issquarefree(d)*d);
%Y A293228 Cf. A005117, A008966, A048250, A293227, A293235.
%K A293228 nonn
%O A293228 1,4
%A A293228 _Antti Karttunen_, Oct 08 2017