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%I #22 Jun 15 2018 05:48:31
%S 24,36,40,48,60,72,80,84,90,96,108,112,120,126,132,140,144,156,160,
%T 168,176,180,192,200,204,208,210,216,224,228,240,252,264,270,276,280,
%U 288,300,312,320,324,336,348,352,360,372,378,384,392,396,400,408,416,420
%N Numbers n such that sigma(n)^2 > 2*sigma(n^2).
%C For every n>1 sigma(n)^2 > sigma(n^2).
%C Lim_{n->inf} a(n)/n appears to exist and is near 8.0; e.g., a(124094) = 1000000. - _Paul D. Hanna_, Sep 22 2011
%C We also have a(12438441) = 10^8, a(124240921) = 10^9, and a(1242729194) = 10^10. - _Giovanni Resta_, Jun 15 2018
%H Paul D. Hanna, <a href="/A067807/b067807.txt">Table of n, a(n) for n = 1..1000</a>
%e The limit a(n)/n seems to be near 8.0:
%e n a(n) a(n)/n
%e ------- -------- ----------
%e 124094 1000000 8.05840...
%e 248310 2000000 8.05444...
%e 372503 3000000 8.05362...
%e 496826 4000000 8.05110...
%e 621163 5000000 8.04941...
%e 745602 6000000 8.04718...
%e 870189 7000000 8.04422...
%e 994799 8000000 8.04182...
%e 1119336 9000000 8.04048...
%e 1243884 10000000 8.03933...
%t Select[Range[500],DivisorSigma[1,#]^2>2DivisorSigma[1,#^2]&] (* _Harvey P. Dale_, Mar 30 2011 *)
%o (PARI) {for(n=1,8000,if(2*sigma(n^2)-sigma(n)^2 < 0,print1(n,",")))} \\ _Paul D. Hanna_, Sep 22 2011
%Y Cf. A195735, A065764.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Feb 07 2002