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%I #26 Apr 29 2019 12:25:11
%S 720,1080,1440,1680,1800,2016,2160,2520,2880,3024,3240,3360,3600,3780,
%T 3960,4032,4200,4320,4680,5040,5280,5400,5544,5760,6048,6120,6300,
%U 6480,6720,6840,7056,7200,7560,7920,8064,8400,8640,9000,9072,9240,9360,9504
%N Numbers k such that sigma(k)^2 > 3*sigma(k^2).
%C For every m>1 sigma(m)^2 > sigma(m^2).
%C From _Robert Israel_, Jun 20 2018: (Start)
%C Numbers with prime factorization Product_j p_j^(e_j) such that Product_j (p_j^(e_j+1)-1)^2/((p_j^(2*e_j+1)-1)*(p_j-1)) > 3.
%C If h is a member then so are all multiples of h.
%C The first member that is squarefree is 7420738134810 = A002110(12).
%C (End)
%H Robert Israel, <a href="/A067808/b067808.txt">Table of n, a(n) for n = 1..10000</a>
%p filter:= proc(n) local F;
%p F:= ifactors(n)[2];
%p mul((t[1]^(t[2]+1)-1)^2/(t[1]^(2*t[2]+1)-1)/(t[1]-1), t = F) > 3
%p end proc:
%p select(filter, [$1..10^4]); # _Robert Israel_, Jun 20 2018
%t filterQ[n_] := Module[{F = FactorInteger[n]}, If[n == 1, Return[False]]; Product[{p, e} = pe; (p^(e+1)-1)^2/((p^(2e+1)-1)(p-1)), {pe, F}] > 3];
%t Select[Range[10^4], filterQ] (* _Jean-François Alcover_, Apr 29 2019, after _Robert Israel_ *)
%o (PARI) isok(k) = sigma(k)^2 > 3*sigma(k^2); \\ _Michel Marcus_, Apr 29 2019
%Y Cf. A002110, A065764.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Feb 07 2002
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