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
KEYWORD
nonn
AUTHOR
Jayanta Basu, Mar 31 2013
STATUS
approved