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

1,2

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

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.

Example

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.

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 *)

Prog

(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

Crossrefs

Intersection of A003601, A020491, and A033950.

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

Sequence in context: A225324 A225358 A115620 * A234114 A234107 A136547

Adjacent sequences: A224105 A224106 A224107 * A224109 A224110 A224111

Keyword

nonn

Author

Jayanta Basu, Mar 31 2013

Status

approved