A066831 - OEIS

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Other ways to Give Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A066831 Numbers k such that sigma(k) divides sigma(phi(k)). 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) isok(k) = { sigma(eulerphi(k)) % sigma(k) == 0 } \\ Harry J. Smith, Mar 30 2010 CROSSREFS Cf. A033631, A067382, A067383, A067384, A067385. Sequence in context: A235454 A296831 A031442 * A067382 A340842 A253776 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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