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A066831 Numbers n such that sigma(n) divides sigma(phi(n)). 7
1, 13, 71, 87, 89, 181, 203, 305, 319, 362, 667, 899, 1257, 1363, 1421, 1525, 1711, 1798, 1889, 2407, 2501, 2933, 3103, 4609, 4615, 4687, 4843, 5002, 5191, 6583, 7123, 7625, 7627, 9374, 9947, 10063, 10411, 10991, 11107, 12989, 13543, 13891, 14587 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For odd n, if sigma(phi(n))/sigma(n)=3 then sigma(phi(2*n))/sigma(2*n)=1. - Vladeta Jovovic, Jan 21 2002.

Comments from Vim Wenders, Nov 01 2006: (Start)

This is almost certainly false for even n. For odd n we have phi(n)=phi(2n) and with sigma(2)=3 trivially sigma(phi(n))/sigma(n)=3 <=> sigma(phi(2n))/sigma(2n) = sigma(phi(n))/3.sigma(n)=1.

But suppose n=2m, m odd: again with phi(2m)=phi(m) and sigma(2)=3, sigma(phi(2m)) / sigma(2m)=3 => sigma(phi( m)) /3sigma( m)=3 => sigma(phi( m)) / sigma( m)=9; and with sigma(4)=7 sigma( phi(4m))/ sigma(4m)=1 => sigma(2phi( m))/7sigma( m)=1 => sigma(2phi( m))/ sigma( m)=7. So we get the condition sigma(phi( m)) / sigma( m)=9 <=> sigma(2phi( m))/ sigma( m)=7 which will fail. So if there is a (very) big odd number n in A066831 (numbers n such that sigma(n) divides sigma(phi(n))) with A066831(n) = 9, the conjecture is wrong. I admit I could not yet find such a number, nor do i really know it exists, i.e., A067385(9) exists. (End)

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B42.

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000

MATHEMATICA

For[ n=1, True, n++, If[ Mod[ DivisorSigma[ 1, EulerPhi[ n ] ], DivisorSigma[ 1, n ] ]==0, Print[ n ] ] ]

Select[Range[15000], Divisible[DivisorSigma[1, EulerPhi[#]], DivisorSigma[1, #]]&] (* Harvey P. Dale, Oct 19 2011 *)

PROG

(PARI) { n=0; for (m=1, 10^10, if (sigma(eulerphi(m)) % sigma(m) == 0, write("b066831.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Mar 30 2010]

CROSSREFS

Cf. A033631, A067382, A067383, A067384, A067385.

Sequence in context: A050403 A235454 A031442 * A067382 A253776 A158941

Adjacent sequences:  A066828 A066829 A066830 * A066832 A066833 A066834

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Jan 19 2002

EXTENSIONS

More terms from Vladeta Jovovic and Robert G. Wilson v, Jan 20 2002

Edited by Dean Hickerson, Jan 20 2002

STATUS

approved

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Last modified April 28 23:30 EDT 2017. Contains 285604 sequences.