A246802 - OEIS

%I #19 Sep 11 2015 14:57:38

%S 92,368,712,1472,94208,1507328,6029312,37412864,24696061952

%N Solutions of sigma(sigma(x)) - phi(phi(x)) = 5x.

%C Theorem: If 2^p-1 is a Mersenne prime greater than 3, then

%C x = 23*2^(p-1) is a solution to the equation,

%C sigma(sigma(x)) - phi(phi(x)) = 5x.

%C Proof:  sigma(sigma(x))- phi(phi(x))

%C       = sigma(sigma(23*2^(p-1))) - phi(phi(23*2^(p-1)))

%C       = sigma(24*(2^p-1)) - phi(22*2^(p-2))

%C       = 60*2^p - 10*2^(p-2)

%C       = 115*2^(p-1)

%C       = 5*x.

%C The multiplicative property of both functions phi and sigma is applied along with the assumption p>2.

%C Note that 712 and 37412864 are two terms of the sequence which are not of the form mentioned in the theorem.

%C a(10) > 10^11. - _Hiroaki Yamanouchi_, Sep 11 2015

%t Do[If[DivisorSigma[1,DivisorSigma[1,n]]-EulerPhi[EulerPhi[n]]==5n, Print[n]],{n,230000000}]

%o (PARI) is(n)=sigma(sigma(n)) - eulerphi(eulerphi(n)) == 5*n \\ _Anders Hellström_, Sep 11 2015

%Y Cf. A000010, A000203, A000668, A137600, A244449, A246630.

%K nonn,more

%O 1,1

%A _Jahangeer Kholdi_ and _Farideh Firoozbakht_, Sep 17 2014

%E a(9) from _Hiroaki Yamanouchi_, Sep 11 2015