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A293235 a(n) is the sum of proper divisors of n that are square.

%I A293235 #19 Dec 29 2023 13:12:34
%S A293235 0,1,1,1,1,1,1,5,1,1,1,5,1,1,1,5,1,10,1,5,1,1,1,5,1,1,10,5,1,1,1,21,1,
%T A293235 1,1,14,1,1,1,5,1,1,1,5,10,1,1,21,1,26,1,5,1,10,1,5,1,1,1,5,1,1,10,21,
%U A293235 1,1,1,5,1,1,1,50,1,1,26,5,1,1,1,21,10,1,1,5,1,1,1,5,1,10,1,5,1,1,1,21,1,50,10,30,1,1,1,5,1
%N A293235 a(n) is the sum of proper divisors of n that are square.
%C A293235 a(n) = 1 if and only if n > 1 is squarefree or the square of a prime. - _Robert Israel_, Oct 08 2017
%H A293235 Antti Karttunen, <a href="/A293235/b293235.txt">Table of n, a(n) for n = 1..16384</a>
%F A293235 a(n) = Sum_{d|n, d<n} A010052(d)*d.
%F A293235 a(n) = A035316(n) - (A010052(n)*n).
%F A293235 G.f.: Sum_{k>=1} k^2 * x^(2*k^2) / (1 - x^(k^2)). - _Ilya Gutkovskiy_, Apr 13 2021
%F A293235 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3/2)-1)/3 = 0.537458449561... . - _Amiram Eldar_, Dec 01 2023
%p A293235 A035316:= n -> mul((p[1]^(p[2]+2-(p[2] mod 2))-1)/(p[1]^2-1), p = ifactors(n)[2]):
%p A293235 f:= n -> A035316(n) - `if`(issqr(n),n,0):
%p A293235 map(f, [$1..100]); # _Robert Israel_, Oct 08 2017
%t A293235 Table[Total[Select[Most[Divisors[n]],IntegerQ[Sqrt[#]]&]],{n,120}] (* _Harvey P. Dale_, Dec 29 2023 *)
%o A293235 (PARI) A293235(n) = sumdiv(n,d,(d<n)*ispower(d,2)*d);
%Y A293235 Cf. A010052, A035316, A078434, A293228, A293234.
%K A293235 nonn
%O A293235 1,8
%A A293235 _Antti Karttunen_, Oct 08 2017