A046642 Numbers k such that k and number of divisors d(k) are relatively prime.

1, 3, 4, 5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Author

Labos Elemer

Keywords

• nonn
• nice
• easy

Comments

Numbers k such that tau(k)^phi(k) == 1 (mod k), where tau(k) is the number of divisors of k (A000005) and phi(k) is the Euler phi function (A000010). - Michel Lagneau, Nov 20 2012

Density is at least 4/Pi^2 = 0.405... since A056911 is a subsequence, and at most 1/2 since all even numbers in this sequence are squares. The true value seems to be around 0.4504. - Charles R Greathouse IV, Mar 27 2013

They are called anti-tau numbers by Zelinsky (see link) and their density is at least 3/Pi^2 (theorem 57 page 15). - Michel Marcus, May 31 2015

From Amiram Eldar, Feb 21 2021: (Start)

Spiro (1981) proved that the number of terms of this sequence that do not exceed x is c * x + O(sqrt(x)*log(x)^3), where 0 < c < 1 is the asymptotic density of this sequence.

The odd numbers whose number of divisors is a power of 2 (the odd terms of A036537) are terms of this sequence. Their asymptotic density is A327839/A076214 = 0.4212451116... which is a better lower bound than 4/Pi^2 for the asymptotic density of this sequence.

A better upper limit than 0.5 can be obtained by considering the subsequence of odd numbers whose 3-adic valuation is not of the form 3*k-1 (i.e., odd numbers without those k with gcd(k, tau(k)) = 3), whose asymptotic density is 6/13 = 0.46153...

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 49, 459, 4535, 45145, 450710, 4504999, 45043234, 450411577, 4504050401, ... (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Mart Abel, Helena Lauer, and Ellen Redi, About the number of τ-numbers relative to polynomials with integer coefficients, Acta Comment. Univ. Tartu. Math. 25, No. 1, 107-117, 2021.
• Claudia A. Spiro, The Frequency with Which an Integral-Valued, Prime-Independent, Multiplicative or Additive Function of n Divides a Polynomial Function of n, Ph. D. Thesis, University of Illinois, Urbana-Champaign, 1981.
• 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

Cf. A000005, A000010, A009191, A009230, A036537, A056911, A076214, A327839.

Programs

Haskell

a046642 n = a046642_list !! (n-1)
a046642_list = map (+ 1) $ elemIndices 1 a009191_list
-- Reinhard Zumkeller, Aug 14 2011

Mathematica

Select[ Range[200], CoprimeQ[#, DivisorSigma[0, #]] &] (* Jean-François Alcover, Oct 20 2011 *)

PARI

is(n)=gcd(numdiv(n),n)==1 \\ Charles R Greathouse IV, Mar 27 2013

Formula

A009191(a(n)) = 1.