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Numbers n such that sigma(n) = 3*phi(n).
7

%I #33 Jan 09 2025 00:25:38

%S 2,15,357,3339,5049,10659,12441,24969,99693,124355,132957,145145,

%T 353133,423657,596037,655707,734517,745503,894387,1406427,1641783,

%U 1823877,1936557,3295047,4108401,4194183,4776201,5574699,5842137,5971251,6132789,6953765,7649915

%N Numbers n such that sigma(n) = 3*phi(n).

%C From _Farideh Firoozbakht_, May 01 2009: (Start)

%C If m>1 and 2*3^m-1 is prime then n = 7*3^(m-1)*(2*3^m-1) is in the sequence.

%C Because sigma(n) = 8*(3^m-1)/2*(2*3^m) = 8*3^m*(3^m-1) = 3*6*(2*3^(m-2))*(2*3^m-2) = 3*phi(7)*phi(3^(m-1))*phi(2*3^m-1) = 3*phi(7*3^(m-1)*(2*3^m-1)) = 3*phi(n). (End)

%H Amiram Eldar, <a href="/A068391/b068391.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.

%t Select[Range[765*10^4],DivisorSigma[1,#]==3EulerPhi[#]&] (* _Harvey P. Dale_, Aug 25 2019 *)

%o (PARI) for(n=1,500000, if(sigma(n)==3*eulerphi(n),print1(n,",")))

%Y Cf. A000203.

%Y Subsequence of A087943 (sigma(k) is a multiple of 3).

%K easy,nonn

%O 1,1

%A _Benoit Cloitre_, Mar 03 2002

%E More terms from _Rick L. Shepherd_, May 14 2002