A246630 Numbers n such that phi(phi(n)) + sigma(sigma(n))=6*n.

344, 4016, 16064, 39208, 69430, 130250, 1028096, 1210928, 4843712, 16449536, 65798144, 309997568

Offset

1,1

Comments

Theorem: If q=2^p-1 is a Mersenne prime greater than 7 then n=251*2^(p-1) is in the sequence.

Proof: phi(phi(n))+sigma(sigma(n))

= phi(phi(251*2^(p-1)))+sigma(sigma(251*2^(p-1)))

= phi(125*2^(p-1))+sigma(252*(2^p-1))

= 100*2^(p-2)+sigma(2^2*3^2*7)*2^p

= 25*2^p+7*13*8*2^p

= 753*2^p

= 6*n.

Note that multiplicative property of both functions phi and sigma is utilized along with the assumption p>3.

The first four terms of the sequence of the above form are 4016, 16064, 1028096 and 16449536.

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

a(13) > 2*10^9. - Hiroaki Yamanouchi, Sep 19 2014

251 and 75683 are both primes satisfying phi(phi(p)) + 4*sigma(sigma(p)) = 12*p. - Michel Marcus, Sep 20 2014

Links

Table of n, a(n) for n=1..12.

Mathematica

Do[If[EulerPhi[EulerPhi[n]]+DivisorSigma[1, DivisorSigma[1, n]]==6n, Print[n]], {n, 16500000}]

Prog

(PARI)
for(n=1, 10^9, if(sigma(sigma(n))+eulerphi(eulerphi(n)) == 6*n, print1(n, ", "))) \\ Derek Orr, Sep 19 2014

Crossrefs

Cf. A000010, A000203, A000668, A244449.

Sequence in context: A183680 A183674 A245994 * A250921 A262791 A186935

Adjacent sequences: A246627 A246628 A246629 * A246631 A246632 A246633

Keyword

nonn,more

Author

Jahangeer Kholdi and Farideh Firoozbakht, Sep 16 2014

Extensions

a(11)-a(12) from Hiroaki Yamanouchi, Sep 19 2014

Status

approved