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A246802 Solutions of sigma(sigma(x)) - phi(phi(x)) = 5x. 0
92, 368, 712, 1472, 94208, 1507328, 6029312, 37412864, 24696061952 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 7 05:10 EDT 2021. Contains 343636 sequences. (Running on oeis4.)