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

Author

Farideh Firoozbakht, Apr 01 2005

Keywords

• easy
• nonn

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

Examples

			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).
		

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.

Crossrefs

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

Programs

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 *)

PARI

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