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
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
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