

A224108


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


1



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
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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



