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A224108 Numbers k such that tau(k) divides k, sigma(k) and phi(k).

%I A224108 #25 Feb 20 2021 04:28:15
%S A224108 1,56,184,248,376,504,568,632,672,824,864,1016,1208,1248,1336,1528,
%T A224108 1592,1656,1784,1824,1912,2016,2104,2168,2232,2488,2688,2872,2936,
%U A224108 2976,3064,3360,3384,3448,3512,3552,3704,3832,3896,3968,4024,4128,4284,4320,4792,4856,5048
%N A224108 Numbers k such that tau(k) divides k, sigma(k) and phi(k).
%C A224108 4 | a(n) for n > 1. Natural density 0. - _Charles R Greathouse IV_, Mar 31 2013
%C A224108 Zelinsky (2002) called these terms "rare numbers", and noted that if p and q are distinct primes, not equal to 2,3 or 7, then 672*p*q is a term. He proved that for any k > 0 and for sufficiently large m the number of terms not exceeding m is > k*pi(m), where pi(m) = A000720(m). - _Amiram Eldar_, Feb 20 2021
%H A224108 Charles R Greathouse IV, <a href="/A224108/b224108.txt">Table of n, a(n) for n = 1..10000</a>
%H A224108 Joshua Zelinsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html">Tau Numbers: A Partial Proof of a Conjecture and Other Results</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
%e A224108 56 is in the sequence because 56 has 8 divisors (1, 2, 4, 7, 8, 14, 28, 56), and 8 is a divisor of 56, as well as of sigma(56) = 120 and of phi(56) = 24.
%t A224108 Select[Range[10000], GCD[DivisorSigma[1, #], #, EulerPhi[#], DivisorSigma[0, #]] == DivisorSigma[0, #] &]
%t A224108 Select[Range[5100],AllTrue[{#,DivisorSigma[1,#],EulerPhi[#]}/ DivisorSigma[ 0,#], IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Oct 02 2019 *)
%o A224108 (PARI) is(n)=my(t=numdiv(n)); n%t==0 && sigma(n)%t==0 && eulerphi(n)%t==0 \\ _Charles R Greathouse IV_, Mar 31 2013
%Y A224108 Intersection of A003601, A020491, and A033950.
%Y A224108 Cf. A000005, A000010, A000203, A000720, A069810, A217301.
%K A224108 nonn
%O A224108 1,2
%A A224108 _Jayanta Basu_, Mar 31 2013