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Numbers k such that sigma(k) = 4*phi(k).
20

%I #56 Aug 19 2024 03:17:58

%S 14,105,248,418,1485,3135,3596,3956,4064,5396,8636,20026,23374,25714,

%T 35074,35343,39105,41656,55154,56134,56536,71145,74613,87087,124605,

%U 150195,175305,192855,263055,393104,413655,421005,474548,604012,697851,711988,819772

%N Numbers k such that sigma(k) = 4*phi(k).

%C If 2^p-1 is a prime (Mersenne prime) greater than 3 then 2^(p-2)*(2^p-1) is in the sequence. So for n>1, 2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence. - _Farideh Firoozbakht_, Feb 23 2005

%C Theorem: If m>0, k is integer and p=2^(m+2)+k-1 is a prime number then n=2^m*p is a solution to the equation sigma(x) = 4*phi(x)-k. The previous comment is the special case k=0. - _Farideh Firoozbakht_, Oct 01 2014

%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 88.

%H Amiram Eldar, <a href="/A068390/b068390.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.

%H Farideh Firoozbakht and M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1

%t Select[Range[900000],DivisorSigma[1,#]==4EulerPhi[#]&] (* _Harvey P. Dale_, Nov 29 2013 *)

%o (PARI) for(n=1,300000, if(sigma(n)==4*eulerphi(n),print1(n,",")))

%o (Magma) [n: n in [1..10^6] | SumOfDivisors(n) eq 4*EulerPhi(n)]; // _Vincenzo Librandi_, Sep 25 2017

%Y Cf. A000010, A000203, A079546, A000043, A292422.

%Y Subsequence of A248150 (sigma(k) is divisible by 4).

%K easy,nonn

%O 1,1

%A _Benoit Cloitre_, Mar 03 2002

%E More terms from _Carl Najafi_, Aug 16 2011