A104901 Numbers n such that sigma(n) = 8*phi(n).

42, 594, 744, 1254, 7668, 8680, 10788, 11868, 12192, 14630, 15642, 16188, 25908, 28458, 49842, 60078, 70122, 77142, 105222, 124968, 125860, 138460, 142240, 165462, 168402, 169608, 188860, 201924, 242316, 259160, 302260, 553000, 561906, 700910, 726440

Offset

1,1

Comments

If p>3 and 2^p-1 is prime (a Mersenne prime) then 35*2^(p-2)*(2^p-1) is in the sequence. So 35*2^(A000043-2)*(2^A000043-1) is a subsequence of this sequence.

If p>2 and 2^p-1 is prime (a Mersenne prime) then 3*2^(p-2)*(2^p-1) is in the sequence (the proof is easy). - Farideh Firoozbakht, Dec 23 2007

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)

Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.

Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

Example

p>3, q=2^p-1(q is prime); m=35*2^(p-2)*q so sigma(m)=48*(2^(p-1)-1)*2^p=8*(24*2^(p-3)*(2^p-2))=8*phi(m) hence m is in the sequence.
sigma(553000) = 1497600 = 8*187200 = 8*phi(553000) so 553000 is in the sequence but 553000 is not of the form 35*2^(p-2)*(2^p-1).

Mathematica

Do[If[DivisorSigma[1, m] == 8*EulerPhi[m], Print[m]], {m, 1000000}]
Select[Range[800000], DivisorSigma[1, #]==8*EulerPhi[#]&] (* Harvey P. Dale, Sep 12 2018 *)

Prog

(PARI) is(n)=sigma(n)==8*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

Crossrefs

Cf. A000043, A062699, A068390, A104900, A104902.

Sequence in context: A245874 A293096 A279888 * A091962 A269659 A007746

Adjacent sequences: A104898 A104899 A104900 * A104902 A104903 A104904

Keyword

easy,nonn

Author

Farideh Firoozbakht, Apr 01 2005

Status

approved