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%I #20 Jun 24 2019 04:38:45
%S 18,28,40,54,68,84,99,120,124,184,204,208,220,284,297,315,372,388,423,
%T 424,475,508,552,616,624,660,765,796,852,900,928,940,945,963,964,1012,
%U 1152,1164,1192,1269,1272,1348,1395,1425,1449,1458,1496,1524,1664,1719,1796,1848,1975
%N Numbers k that are not squarefree such that the difference between sigma(k) and usigma(k) is a square > 0.
%C This sequence is a subsequence of A013929 (nonsquarefree numbers).
%C If n were a squarefree number (A005117), then the difference would be 0.
%C It appears that the terms are of the form q*p^q, p prime.
%C The corresponding squares are: 9, 16, 36, 36, 36, 64, 36, 144, 64, 144, 144, 196, 144,....
%H Amiram Eldar, <a href="/A216259/b216259.txt">Table of n, a(n) for n = 1..10000</a>
%p with(numtheory): for n from 1 to 2000 do :it:=1:s:=0:x:=divisors(n): n1:=nops(x): for k from 1 to n1 do:d:=x[k]:if gcd(d,n/d)=1 then s:=s+d:else fi:od: s1:=sigma(n): if sqrt(s1-s)=floor(sqrt(s1-s)) and s1>s then printf(`%d, `,n):else fi:od:
%t lst={}; usigma[n_] := Block[{d=Divisors[n]}, DivisorSigma[1,n] - Plus@@Select[d,GCD[#,n/#] == 1&]]; Do[If[IntegerQ[Sqrt[usigma[n] && usigma[n] > 0]], AppendTo[lst,n]], {n,2000}]; lst
%Y Cf. A000203 (sigma:sum of divisors), A034448 (usigma:sum of unitary divisors).
%Y Cf. A064212 (sigma+usigma), A013929 (nonsquarefree numbers).
%K nonn
%O 1,1
%A _Michel Lagneau_, Mar 15 2013
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