A246802 Solutions of sigma(sigma(x)) - phi(phi(x)) = 5x.

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

Offset

1,1

Comments

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

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

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

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

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

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

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

= 115*2^(p-1)

= 5*x.

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

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

a(10) > 10^11. - Hiroaki Yamanouchi, Sep 11 2015

Links

Table of n, a(n) for n=1..9.

Mathematica

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

Prog

(PARI) is(n)=sigma(sigma(n)) - eulerphi(eulerphi(n)) == 5*n \\ Anders Hellström, Sep 11 2015

Crossrefs

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

Sequence in context: A044805 A063326 A185461 * A302351 A100169 A100170

Adjacent sequences: A246799 A246800 A246801 * A246803 A246804 A246805

Keyword

nonn,more

Author

Jahangeer Kholdi and Farideh Firoozbakht, Sep 17 2014

Extensions

a(9) from Hiroaki Yamanouchi, Sep 11 2015

Status

approved