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%I #26 Mar 02 2015 17:00:45
%S 344,4016,16064,39208,69430,130250,1028096,1210928,4843712,16449536,
%T 65798144,309997568
%N Numbers n such that phi(phi(n)) + sigma(sigma(n))=6*n.
%C Theorem: If q=2^p-1 is a Mersenne prime greater than 7 then n=251*2^(p-1) is in the sequence.
%C Proof: phi(phi(n))+sigma(sigma(n))
%C = phi(phi(251*2^(p-1)))+sigma(sigma(251*2^(p-1)))
%C = phi(125*2^(p-1))+sigma(252*(2^p-1))
%C = 100*2^(p-2)+sigma(2^2*3^2*7)*2^p
%C = 25*2^p+7*13*8*2^p
%C = 753*2^p
%C = 6*n.
%C Note that multiplicative property of both functions phi and sigma is utilized along with the assumption p>3.
%C The first four terms of the sequence of the above form are 4016, 16064, 1028096 and 16449536.
%C If q = 2^p-1 is a Mersenne prime greater than 7 then n = 75683*2^(p-1) is in the sequence. - _Hiroaki Yamanouchi_, Sep 19 2014
%C a(13) > 2*10^9. - _Hiroaki Yamanouchi_, Sep 19 2014
%C 251 and 75683 are both primes satisfying phi(phi(p)) + 4*sigma(sigma(p)) = 12*p. - _Michel Marcus_, Sep 20 2014
%t Do[If[EulerPhi[EulerPhi[n]]+DivisorSigma[1,DivisorSigma[1,n]]==6n,Print[n]],{n,16500000}]
%o (PARI)
%o for(n=1,10^9,if(sigma(sigma(n))+eulerphi(eulerphi(n)) == 6*n,print1(n,", "))) \\ _Derek Orr_, Sep 19 2014
%Y Cf. A000010, A000203, A000668, A244449.
%K nonn,more
%O 1,1
%A _Jahangeer Kholdi_ and _Farideh Firoozbakht_, Sep 16 2014
%E a(11)-a(12) from _Hiroaki Yamanouchi_, Sep 19 2014