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%I #18 Sep 19 2014 10:00:25
%S 20,80,320,20480,65792,327680,1310720
%N Numbers n such that phi(phi(n))+sigma(sigma(n))=5n.
%C Theorem: If q=2^p-1 is a Mersenne prime greater than 3 then n=5*2^(p-1) is in the sequence.
%C Proof: phi(phi(n))+sigma(sigma(n))
%C = phi(phi(5*2^(p-1)))+sigma(sigma(5*2^(p-1)))
%C = phi(4*2^(p-2))+sigma(6*(2^p-1))
%C = 2^(p-1)+12*2^p
%C = 25*(2^(p-1))
%C = 5*n.
%C Note that multiplicative property of both functions phi and sigma is utilized along with the assumption p>2.
%C Perhaps 65792 is the only term of the sequence which is not of this form.
%C a(8) > 10^9. - _Hiroaki Yamanouchi_, Sep 19 2014
%t Select[Range[2000000],EulerPhi[EulerPhi[#]]+DivisorSigma[1,DivisorSigma[1,#]]==5#&]
%o (PARI) isok(n) = eulerphi(eulerphi(n))+sigma(sigma(n)) == 5*n; \\ _Michel Marcus_, Sep 17 2014
%Y Cf. A000010, A000203, A000668, A246630.
%K nonn,more
%O 1,1
%A _Jahangeer Kholdi_ and _Farideh Firoozbakht_, Sep 16 2014
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