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Numbers n such that sigma_2(n) > phi(n)^3.
2

%I #7 Nov 19 2015 17:16:47

%S 2,3,4,6,8,10,12,14,18,20,24,30,36,42,60

%N Numbers n such that sigma_2(n) > phi(n)^3.

%C This sequence is finite, since by Grönwall's theorem sigma_2(n) <= sigma(n)^2 << (n log log n)^2 but phi(n)^3 >> (n/log log n)^3. - _Charles R Greathouse IV_, Nov 18 2015

%F Solutions to A001157(n) > A000010(x)^3.

%e d-square sums:{5, 10, 21, 50, 85, 130, 210, 250, 455, 546, 850, 1300, 1911, 2500, 5460} phi-cubes:{1, 8, 8, 8, 64, 64, 64, 216, 216, 512, 512, 512, 1728, 1728, 4096} differences:{4, 2, 13, 42, 21, 66, 146, 34, 239, 34, 338, 788, 183, 772, 1364} Sequence is believed to be full.

%t Select[Range[100],DivisorSigma[2,#]>EulerPhi[#]^3&] (* _Harvey P. Dale_, Feb 19 2013 *)

%o (PARI) is(n)=my(f=factor(n)); sigma(f,2)>eulerphi(f)^3 \\ _Charles R Greathouse IV_, Nov 18 2015

%Y Cf. A001157, A000010, A055234.

%K nonn,fini

%O 1,1

%A _Labos Elemer_, Sep 27 2001