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A246630
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Numbers n such that phi(phi(n)) + sigma(sigma(n))=6*n.
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2
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344, 4016, 16064, 39208, 69430, 130250, 1028096, 1210928, 4843712, 16449536, 65798144, 309997568
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OFFSET
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1,1
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COMMENTS
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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
251 and 75683 are both primes satisfying phi(phi(p)) + 4*sigma(sigma(p)) = 12*p. - Michel Marcus, Sep 20 2014
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LINKS
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MATHEMATICA
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Do[If[EulerPhi[EulerPhi[n]]+DivisorSigma[1, DivisorSigma[1, n]]==6n, Print[n]], {n, 16500000}]
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PROG
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(PARI)
for(n=1, 10^9, if(sigma(sigma(n))+eulerphi(eulerphi(n)) == 6*n, print1(n, ", "))) \\ Derek Orr, Sep 19 2014
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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