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

1, 56, 184, 248, 376, 504, 568, 632, 672, 824, 864, 1016, 1208, 1248, 1336, 1528, 1592, 1656, 1784, 1824, 1912, 2016, 2104, 2168, 2232, 2488, 2688, 2872, 2936, 2976, 3064, 3360, 3384, 3448, 3512, 3552, 3704, 3832, 3896, 3968, 4024, 4128, 4284, 4320, 4792, 4856, 5048

Offset: 1

Author

Jayanta Basu, Mar 31 2013

Keywords

• nonn

Comments

4 | a(n) for n > 1. Natural density 0. - Charles R Greathouse IV, Mar 31 2013

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

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Crossrefs

Intersection of A003601, A020491, and A033950.

Cf. A000005, A000010, A000203, A000720, A069810, A217301.

Programs

Mathematica

Select[Range[10000], GCD[DivisorSigma[1, #], #, EulerPhi[#], DivisorSigma[0, #]] == DivisorSigma[0, #] &]
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 *)

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