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%I #32 Dec 04 2019 18:18:51
%S 42,594,744,1254,7668,8680,10788,11868,12192,14630,15642,16188,25908,
%T 28458,49842,60078,70122,77142,105222,124968,125860,138460,142240,
%U 165462,168402,169608,188860,201924,242316,259160,302260,553000,561906,700910,726440
%N Numbers n such that sigma(n) = 8*phi(n).
%C 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.
%C 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
%H Amiram Eldar, <a href="/A104901/b104901.txt">Table of n, a(n) for n = 1..10000</a> (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
%H Kevin A. Broughan and Daniel Delbourgo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Broughan/broughan26.html">On the Ratio of the Sum of Divisors and Euler’s Totient Function I</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
%H Kevin A. Broughan and Qizhi Zhou, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Broughan/bro32.html">On the Ratio of the Sum of Divisors and Euler's Totient Function II</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.
%e 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.
%e 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).
%t Do[If[DivisorSigma[1, m] == 8*EulerPhi[m], Print[m]], {m, 1000000}]
%t Select[Range[800000],DivisorSigma[1,#]==8*EulerPhi[#]&] (* _Harvey P. Dale_, Sep 12 2018 *)
%o (PARI) is(n)=sigma(n)==8*eulerphi(n) \\ _Charles R Greathouse IV_, May 09 2013
%Y Cf. A000043, A062699, A068390, A104900, A104902.
%K easy,nonn
%O 1,1
%A _Farideh Firoozbakht_, Apr 01 2005
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